section 4.1 the product, quotient, and power rules for exponents
TRANSCRIPT
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Section 4.1
The Product, Quotient, and Power Rules for Exponents
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OBJECTIVES
Multiply expressions using the product rule for exponents.
A
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OBJECTIVES
Divide expressions using the quotient rule for exponents.
B
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OBJECTIVES
Use the power rules to simplify expressions.
C
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RULESSigns for Multiplication
1. When multiplying two numbers with the same sign, product is positive (+).
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RULESSigns for Multiplication
2. When multiplying two numbers with different signs, product is negative (-).
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RULESSigns for Division
1.When dividing two numbers with the same sign, product is positive (+).
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RULESSigns for Division
2.When dividing two numbers with different signs, product is negative (-).
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RULES FOR EXPONENTSIf m, n, and k are positive integers, then:1. Product rule for exponents
xmxn = xm+n
Example:
x5•x6 = x5+6 = x11
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RULES FOR EXPONENTSIf m, n, and k are positive integers, then:2. Quotient rule for exponents
- > , 0=m m nn m n xx x
x
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RULES FOR EXPONENTSIf m, n, and k are positive integers, then:2. Quotient rule for exponents
Example:
p8
p3 = p8-3 = p5
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RULES FOR EXPONENTSIf m, n, and k are positive integers, then:
3. Power rule for products
=k mk nkm n yy xx
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RULES FOR EXPONENTSIf m, n, and k are positive integers, then:
3. Power rule for products
Example:
= =4
4 3 4 4 3 4 16 12x y x xy y• •
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RULES FOR EXPONENTSIf m, n, and k are positive integers, then:
4. Power rule for quotients
0=m m
m yx x y y
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RULES FOR EXPONENTSIf m, n, and k are positive integers, then:
4. Power rule for quotients Example:
6
= =3 3 6 184 4 6 24
a a ab b b
••
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Section 4.1Exercise #1
Chapter 4Exponents and Polynomials
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Find.
a. (2a3b)(– 6ab3 )
= (2 • – 6)a3+1 b1+3
= – 12a4b4
b. (– 2x 2yz)(– 6xy3z 4)
= ( – 2 • – 6)x 2 + 1 y1 + 3 z1 + 4
= 12x3y5z5
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Find.
c. 18x5y7
– 9xy3
= 18
– 9
x5 – 1 y7 – 3
= – 2x 4y 4
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Section 4.1Exercise #2
Chapter 4Exponents and Polynomials
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Find.
3 2 3 3 3 3 2 3(2 ) = 2 x y x y
= 8x 9y6
b. ( – 3x 2y3 )2
a. (2x3y 2 )3
2 3 2 2 2 2 3 2( – 3 ) = ( – 3) x y x y
= 9x 4y6
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Section 4.2
Integer Exponents
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OBJECTIVES
Write an expression with negative exponents as an equivalent one with positive exponents.
A
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OBJECTIVES
Write a fraction involving exponents as a number with a negative power.
B
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OBJECTIVES
Multiply and divide expressions involving negative exponents.
C
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RULESZero Exponent
0For 0, =1x x
– n 1= 0nx xx
If n is a positive integer,Negative Exponent
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RULESnth Power of a Quotient
–1 =
nnxx
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RULES
x–m
y–n = yn
xm
For any nonzero numbers x and y and any positive integers m and n:
Simplifying Fractions with Negative Exponents
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Section 4.2Exercise #4
Chapter 4Exponents and Polynomials
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Simplify and write the answer without negative exponents.
– 71a. x
– 7– 1 = x
= x( – 1) ( – 7 )
= x 7
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Simplify and write the answer without negative exponents.
b. x – 6
x – 6
= x – 6 – – 6
0 = = 1, 0xx
= x – 6 + 6
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Section 4.2Exercise #5
Chapter 4Exponents and Polynomials
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Simplify.– 3 4
2 3
– 2
2
b. 3
x yx y
= 2 –2 x – 3 –2 y 4 –2
3–2 x
2 –2 y
3 –2
= 2 –2 x 6 y –8
3–2 x
– 4 y
–6
=
32 x 6 – – 4 y –8 –(–6)
22
=
9 x10 y –2
4
= 9 x10
4y2
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Simplify.
= 2 – 2 3 – 1( – 2) x – 5( – 2) y ( – 2)
= 2 – 2 3 2 x 10 y – 2
2 102 21 1 = 3
2 x
y
= 9x10
4y2
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Section 4.3
Applicationof Exponents:Scientific Notation
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OBJECTIVES
Write numbers in scientific notation.
A
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OBJECTIVES
Multiply and divide numbers in scientific notation.
B
Solve applications.C
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RULES
M10n
A number in scientific notation is written as
Where M is a number between 1 and 10 and n is an integer.
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PROCEDURE
1. Move decimal point in number so there is only one nonzero digit to its left.
(M10n)
The resulting number is M.
Writing a number in scientific notation
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PROCEDURE
2. If the decimal point is moved to the left, n is positive;
(M10n)Writing a number in scientific notation
If the decimal point is moved to the right, n is negative.
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PROCEDURE
3. Write (M10n).
(M10n)Writing a number in scientific notation
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PROCEDUREMultiplying using scientific notation
1. Multiply decimal parts first. Write result in scientific notation.
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PROCEDUREMultiplying using scientific notation
2. Multiply powers of 10 using product rule.
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PROCEDUREMultiplying using scientific notation
3. Answer is product obtained in steps 1 and 2 after simplification.
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Section 4.3Exercise #6
Chapter 4Exponents and Polynomials
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a. 48,000,000
Write in scientific notation.
= 4 8000000 .
= 4.8107
b. 0.00000037
= 0.0000003 7
= 3.7 10 – 7
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Section 4.3Exercise #7
Chapter 4Exponents and Polynomials
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Perform the indicated operations.
4 6a. 3 10 7.1 10
4 + 6 = 3 7.1 10
= 21.3 1010
= 2.13 101 + 10
= 2.13 1011
= 2.13 101 1010
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Section 4.4
Polynomials:An Introduction
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OBJECTIVES
Classify polynomials.A
Find the degree of a polynomial.
B
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OBJECTIVES
Write a polynomial in descending order.
C
Evaluate polynomials.D
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DEFINITIONPolynomialAn algebraic expression formed using addition and subtraction on products of numbers and variables raised to whole number exponents.
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Section 4.4Exercise #8
Chapter 4Exponents and Polynomials
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Classify as a monomial (M), binomial (B), or trinomial (T).
a. 3x – 5
B, binomial
b. 5x3
M, monomial
c. 8x 2 – 2 + 5x
T, trinomial
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Section 4.4Exercise #10
Chapter 4Exponents and Polynomials
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Find the value.
– 16t 2 + 100 when t = 2
= – 16(2)2 + 100
= – 16(4) + 100
= – 64 + 100
= 36
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Section 4.5
Addition and Subtraction of Polynomials
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OBJECTIVES
Add polynomials.A
Subtract polynomials.B
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OBJECTIVES
Find areas by adding polynomials.
C
Solve applications.D
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Section 4.5Exercise #11
Chapter 4Exponents and Polynomials
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Add.
2 – 4 + 8 – 3 + –5 – 4 + 2 2x x x x
= – 4x + 8x 2 – 3 – 5x 2 – 4 + 2x
= ( 8x 2 – 5x 2) + ( – 4x + 2x ) + ( – 3 – 4)
= 3x 2 – 2x – 7
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Section 4.5Exercise #12
Chapter 4Exponents and Polynomials
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23 – 2 – 5 – 2 + 82x x x x
= 3x 2 – 2x – 5x + 2 – 8x 2
= (3x 2 – 8x 2) + ( – 2x – 5x ) + 2
= – 5x 2 – 7x +2
Subtract 5x – 2 + 8x 2 from 3x2 – 2x.
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Section 4.6
Multiplicationof Polynomials
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OBJECTIVES
Multiply two monomials.A
Multiply a monomial and a binomial.
B
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OBJECTIVES
Multiply two binomials using FOIL method.
C
Solve an application.D
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PROCEDURE
First terms multiplied first.
FOIL Method for Multiplying Binomials
Outer terms multiplied second.
Inner terms multiplied third.
Last terms multiplied last.
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Section 4.6Exercise #16
Chapter 4Exponents and Polynomials
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Find (5x – 2y ) (4x – 3y ) .
= 20x 2 – 23xy + 6y 2
= 20x 2 – 15xy – 8xy + 6y 2F O I L
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Section 4.7
Special Productof Polynomials
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OBJECTIVES
Expand binomials of the form
A (X +A)2
B (X – A)2
C (X +A)(X – A)
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OBJECTIVES
Multiply a binomial by a trinomial.
D
Multiply any two polynomials.
E
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SPECIAL PRODUCTS
(X +A)(X +B)= X 2+(A+B)X +AB
SP1 or FOIL
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SPECIAL PRODUCTS
SP2
(X +A)(X +A)=(X +A)2
= X 2+2AX +A2
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SPECIAL PRODUCTS
SP3
(X -A)(X -A)=(X -A)2
= X 2 -2AX +A2
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SPECIAL PRODUCTS
2 2( + )( - )= -X A X A X A
SP4
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PROCEDUREMultiplying Any Two Polynomials (Term-By-Term Multiplication)
Multiply each term of one by every term of other and add results.
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PROCEDUREAppropriate Method for Multiplying Two Polynomials:1. Is the product the square
of a binomial?
Both answers have three terms.
If so, use SP2 or SP3.
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PROCEDUREAppropriate Method for Multiplying Two Polynomials:2. Are the two binomials in the
product the sum and difference of the same two terms?
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PROCEDUREAppropriate Method for Multiplying Two Polynomials:
Answer has two terms.
If so, use SP4.
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PROCEDUREAppropriate Method for Multiplying Two Polynomials:
3. Is the binomial product different from previous two?
Answer has three or four terms.If so, use FOIL.
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PROCEDUREAppropriate Method for Multiplying Two Polynomials:
4. Is product still different? If so, multiply every term of first polynomial by every term of second and collect like terms.
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Section 4.7Exercise #18
Chapter 4Exponents and Polynomials
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Expand.
(2x – 7y )2 (a – b)
2 = a
2– 2 ab + b
2
= 4x 2 – 28xy + 49y 2
= (2x)2
– 2 (2x)(7y) + ( 7y ) 2
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Section 4.7Exercise #19
Chapter 4Exponents and Polynomials
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Find (2x – 5y )(2x + 5y).
= (2x )2 – (5y )2
= 4x 2 – 25y 2
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Section 4.7Exercise #20
Chapter 4Exponents and Polynomials
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Find (x + 2)(x2 + 5x + 3)
= x (x2 + 5x + 3) + 2(x2 + 5x + 3)
= x 3 + 5x 2 + 3x + 2x 2 + 10x + 6
= x 3 + (5x 2 + 2x 2 ) + (3x + 10x ) + 6
= x 3 + 7x 2 + 13x + 6
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Section 4.8
Divisionof Polynomials
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OBJECTIVES
Divide a polynomial by a monomial.
A
Divide one polynomial by another polynomial.
B
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RULETo Divide A Polynomial By A Monomial
Divide each term in polynomial by monomial.
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Section 4.8Exercise #25
Chapter 4Exponents and Polynomials
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x – 2 2x3 + 0x 2 – 9x + 5
2x3 – 4x 2
4x 2 – 9x + 5
4x 2 – 8x – 1x + 5
– 1x + 2 3
2x 2 + 4x – 1 R 3
Divide.
2x3 – 9x + 5 by x – 2
Remainder