46polynomial expressions
TRANSCRIPT
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Polynomial Expressions
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A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
Polynomial Expressions
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Example A.2 + 3x
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
Polynomial Expressions
![Page 4: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/4.jpg)
Example A.2 + 3x “the sum of 2 and 3 times x”
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
Polynomial Expressions
![Page 5: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/5.jpg)
Example A.2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
Polynomial Expressions
![Page 6: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/6.jpg)
Example A.2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x”
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
Polynomial Expressions
![Page 7: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/7.jpg)
Example A.2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
Polynomial Expressions
![Page 8: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/8.jpg)
Example A.2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
Polynomial Expressions
![Page 9: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/9.jpg)
Example A.2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
An expression of the form #xN, where the exponent N is a non-negative integer and # is a number, is called a monomial (one-term).
Polynomial Expressions
![Page 10: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/10.jpg)
Example A.2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
An expression of the form #xN, where the exponent N is a non-negative integer and # is a number, is called a monomial (one-term). For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions
![Page 11: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/11.jpg)
Example A.2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = –4 a. 3y2
An expression of the form #xN, where the exponent N is a non-negative integer and # is a number, is called a monomial (one-term). For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions
![Page 12: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/12.jpg)
Example A.2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = –4 a. 3y2 3y2 3(–4)2
An expression of the form #xN, where the exponent N is a non-negative integer and # is a number, is called a monomial (one-term). For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions
![Page 13: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/13.jpg)
Example A.2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”
A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.
Example B. Evaluate the monomials if y = –4 a. 3y2 3y2 3(–4)2 = 3(16) = 48
An expression of the form #xN, where the exponent N is a non-negative integer and # is a number, is called a monomial (one-term). For example, 3x2, –4x3, and 5x6 are monomials.
Polynomial Expressions
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b. –3y2 (y = –4)Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
c. –3y3
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3 = – 3(–64)
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3 = – 3(–64) = 192
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3 = – 3(–64) = 192
Polynomial Expressions
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3 = – 3(–64) = 192
The sum of monomials are called polynomials (many-terms), these are expressions of the form #xN ± #xN-1 ± … ± #x1 ± #where # can be any number.
Polynomial Expressions
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3 = – 3(–64) = 192
The sum of monomials are called polynomials (many-terms), these are expressions of the form #xN ± #xN-1 ± … ± #x1 ± #where # can be any number.
For example, 4x + 7,
Polynomial Expressions
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3 = – 3(–64) = 192
The sum of monomials are called polynomials (many-terms), these are expressions of the form #xN ± #xN-1 ± … ± #x1 ± #where # can be any number.
For example, 4x + 7, –3x2 – 4x + 7,
Polynomial Expressions
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3 = – 3(–64) = 192
The sum of monomials are called polynomials (many-terms), these are expressions of the form #xN ± #xN-1 ± … ± #x1 ± #where # can be any number.
For example, 4x + 7, –3x2 – 4x + 7, –5x4 + 1 are polynomials,
Polynomial Expressions
Polynomial Expressions
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b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.
c. –3y3
–3y3 – 3(–4)3 = – 3(–64) = 192
The sum of monomials are called polynomials (many-terms), these are expressions of the form #xN ± #xN-1 ± … ± #x1 ± #where # can be any number.
For example, 4x + 7, –3x2 – 4x + 7, –5x4 + 1 are polynomials,
x1 is not a polynomial.whereas the expression
Polynomial Expressions
Polynomial Expressions
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Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.Polynomial Expressions
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Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.The polynomial 4x2 – 3x3
is the combination of two monomials; 4x2 and –3x3.
Polynomial Expressions
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Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.The polynomial 4x2 – 3x3
is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.
Polynomial Expressions
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Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.The polynomial 4x2 – 3x3
is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression,
Polynomial Expressions
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Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.The polynomial 4x2 – 3x3
is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3
Polynomial Expressions
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Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.The polynomial 4x2 – 3x3
is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3
= 4(9) – 3(–27)
Polynomial Expressions
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Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.The polynomial 4x2 – 3x3
is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3
= 4(9) – 3(–27)= 36 + 81 = 117
Polynomial Expressions
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Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.The polynomial 4x2 – 3x3
is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3
= 4(9) – 3(–27)= 36 + 81 = 117 Given a polynomial, each monomial is called a term.
Polynomial Expressions
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Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.The polynomial 4x2 – 3x3
is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3
= 4(9) – 3(–27)= 36 + 81 = 117 Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± #
terms
Polynomial Expressions
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Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.The polynomial 4x2 – 3x3
is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3
= 4(9) – 3(–27)= 36 + 81 = 117 Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± #
termsTherefore the polynomial –3x2 – 4x + 7 has 3 terms, –3x2 , –4x and + 7.
Polynomial Expressions
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Each term is addressed by the variable part. Polynomial Expressions
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Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2,
Polynomial Expressions
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Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
Polynomial Expressions
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Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7.
Polynomial Expressions
![Page 41: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/41.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term.
Polynomial Expressions
![Page 42: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/42.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Polynomial Expressions
![Page 43: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/43.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Operations with Polynomials
Polynomial Expressions
![Page 44: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/44.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Operations with Polynomials
Polynomial Expressions
![Page 45: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/45.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined.
Operations with Polynomials
Polynomial Expressions
![Page 46: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/46.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x
Operations with Polynomials
Polynomial Expressions
![Page 47: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/47.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.
Operations with Polynomials
Polynomial Expressions
![Page 48: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/48.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined.
Operations with Polynomials
Polynomial Expressions
![Page 49: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/49.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.
Operations with Polynomials
Polynomial Expressions
![Page 50: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/50.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.
Operations with Polynomials
Polynomial Expressions
![Page 51: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/51.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.When multiplying a number with a term, we multiply it with the coefficient.
Operations with Polynomials
Polynomial Expressions
![Page 52: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/52.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x
Operations with Polynomials
Polynomial Expressions
![Page 53: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/53.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x,
Operations with Polynomials
Polynomial Expressions
![Page 54: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/54.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x.
Operations with Polynomials
Polynomial Expressions
![Page 55: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/55.jpg)
Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .
Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x.
Operations with Polynomials
When multiplying a number with a polynomial, we may expand using the distributive law: A(B ± C) = AB ± AC.
Polynomial Expressions
![Page 56: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/56.jpg)
Example D. Expand and simplify.Polynomial Operations
![Page 57: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/57.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x)
Polynomial Operations
![Page 58: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/58.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12
Polynomial Operations
![Page 59: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/59.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x
Polynomial Operations
![Page 60: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/60.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4
Polynomial Operations
![Page 61: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/61.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)
Polynomial Operations
![Page 62: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/62.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15
Polynomial Operations
![Page 63: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/63.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12
Polynomial Operations
![Page 64: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/64.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3
Polynomial Operations
![Page 65: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/65.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable.
![Page 66: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/66.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example E.
a. (3x2)(2x3) =b. 3x2(–4x) =c. 3x2(2x3 – 4x) =
![Page 67: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/67.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example E.
a. (3x2)(2x3) = 3*2x2x3
b. 3x2(–4x) =c. 3x2(2x3 – 4x) =
![Page 68: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/68.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) =c. 3x2(2x3 – 4x) =
![Page 69: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/69.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) = 3(–4)x2x = –12x3 c. 3x2(2x3 – 4x) =
![Page 70: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/70.jpg)
Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3
Polynomial Operations
When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example E.
a. (3x2)(2x3) = 3*2x2x3 = 6x5
b. 3x2(–4x) = 3(–4)x2x = –12x3 c. 3x2(2x3 – 4x) distribute = 6x5 – 12x3
![Page 71: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/71.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
![Page 72: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/72.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.a. (3x + 2)(2x – 1)
![Page 73: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/73.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
= 3x(2x – 1) + 2(2x – 1)a. (3x + 2)(2x – 1)
![Page 74: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/74.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2
a. (3x + 2)(2x – 1)
![Page 75: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/75.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2
a. (3x + 2)(2x – 1)
![Page 76: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/76.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2
a. (3x + 2)(2x – 1)
![Page 77: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/77.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)
a. (3x + 2)(2x – 1)
![Page 78: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/78.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)= 4x3 + 6x2 – 8x – 2x2 – 3x + 4
a. (3x + 2)(2x – 1)
![Page 79: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/79.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)= 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
![Page 80: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/80.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)= 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.)
![Page 81: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/81.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)= 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.) Fact. If P and Q are two polynomials then PQ ≡ QP.
![Page 82: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/82.jpg)
To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.
Polynomial Operations
Example F.
b. (2x – 1)(2x2 + 3x –4)
= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2
= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)= 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4
a. (3x + 2)(2x – 1)
Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.) Fact. If P and Q are two polynomials then PQ ≡ QP. A shorter way to multiply is to bypass the 2nd step and use the general distributive law.
![Page 83: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/83.jpg)
General Distributive Rule:Polynomial Operations
![Page 84: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/84.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)
Polynomial Operations
![Page 85: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/85.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..
Polynomial Operations
![Page 86: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/86.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..
Polynomial Operations
![Page 87: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/87.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..
Polynomial Operations
![Page 88: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/88.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
Polynomial Operations
![Page 89: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/89.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2
Polynomial Operations
![Page 90: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/90.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x
Polynomial Operations
![Page 91: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/91.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x
Polynomial Operations
![Page 92: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/92.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12
Polynomial Operations
![Page 93: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/93.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify = x2 – x – 12
Polynomial Operations
![Page 94: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/94.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify = x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
![Page 95: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/95.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify = x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3
![Page 96: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/96.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify = x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2
![Page 97: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/97.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify = x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x
![Page 98: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/98.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify = x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2
![Page 99: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/99.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify = x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x
![Page 100: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/100.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify = x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x + 6
![Page 101: 46polynomial expressions](https://reader035.vdocuments.mx/reader035/viewer/2022081605/58e94ee81a28ab262c8b58f3/html5/thumbnails/101.jpg)
General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..Example G. Expand a. (x + 3)(x – 4)
= x2 – 4x + 3x – 12 simplify = x2 – x – 12
b. (x – 3)(x2 – 2x – 2)
Polynomial Operations
= x3 – 2x2 – 2x – 3x2 + 6x + 6 = x3– 5x2 + 4x + 6
We will address the division operation of polynomials later-after we understand more about the multiplication operation.