chapter 6 exponents and polynomials. § 6.1 adding and subtracting polynomials
TRANSCRIPT
Chapter 6
Exponents and Polynomials
§ 6.1
Adding and Subtracting Polynomials
Blitzer, Introductory Algebra, 5e – Slide #3 Section 6.1
Polynomials
A polynomial is a single term or the sum of two or more terms containing variables with whole number exponents.
Consider the polynomial: 6523 34 xxx
This polynomial contains four terms. It is customary to write the terms in order of descending powers of the variable. This is the standard form of a polynomial. Here are two other polynomials which are written in standard form.
638
82754
23
xx
xxx
Blitzer, Introductory Algebra, 5e – Slide #4 Section 6.1
Polynomials
The degree of a polynomial is the greatest degree of any term of the polynomial. The degree of a term
6523 34 xxx
mn yax is (n +m)
and the coefficient of the term is a. If there is exactly one term of greatest degree, it is called the leading term. It’ s coefficient is called the leading coefficient. Consider the polynomial:
3 is the leading coefficient. The degree is 4.
The Degree of axn • If a ≠ 0, the degree of axn is n. The degree of a nonzero constant
term is 0. The constant 0 has no defined degree.
8275 23 xxx
Degree 3 Degree 2 Degree 1
Degree of nonzero constant: 0
Polynomials
Blitzer, Introductory Algebra, 5e – Slide #5 Section 6.1
• The degree of a polynomial is the degree of its highest order term.
• Example:
Degree 3 Polynomial:
Degree 4 Polynomial: 638
82754
23
xx
xxx
Degree of a Polynomial
Blitzer, Introductory Algebra, 5e – Slide #6 Section 6.1
• Monomial: A polynomial with one term.• Binomial: A polynomial with two terms.• Trinomial: A polynomial with three terms.
Example: This is a 4th degree trinomial.
638 4 xx
Special Polynomials
Blitzer, Introductory Algebra, 5e – Slide #7 Section 6.1
Blitzer, Introductory Algebra, 5e – Slide #8 Section 6.1
Polynomials
The Degree of TIf , the degree of is n. The degree of a nonzero constant is 0. The constant 0 has no defined degree.
nax0a
nax
Adding PolynomialsPolynomials are added by removing the parentheses that surround each polynomial (if any) and then combining like terms.
Subtracting PolynomialsTo subtract two polynomials, change the sign of every term of the second polynomial. Add this result to the first polynomial.
• Polynomials are added by combining like terms. • Like terms are terms containing exactly the same
variables to the same powers.
• Example:2222 10)64(64 xxxx
These like terms both contain x to the second power.
Add the coefficients and keep the same variable
factor, x2
Adding Polynomials
Blitzer, Introductory Algebra, 5e – Slide #9 Section 6.1
518123
)5()18(123
)72()810()48()52(
)7845()21082 :Add
23
23
2233
2323
xxx
xxx
xxxxxx
xxxx-xx(-
Group like terms.
Combine like terms.
Adding Polynomials
Blitzer, Introductory Algebra, 5e – Slide #10 Section 6.1
EXAMPLEEXAMPLE
Blitzer, Introductory Algebra, 5e – Slide #11 Section 6.1
Adding Polynomials
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Add: . 1771119131167 2323 xxxxxx
1771119131167 2323 xxxxxx
1771119131167 2323 xxxxxx Remove parentheses
4 4 5 12
1713711116197
23
2233
xxx
xxxxxx Rearrange terms so that like terms are adjacent
Combine like terms
943x
9)4(3
)72()610()58(
) 765()2108(
)765()2108 :Subtract
2
2
22
22
22
x
xx
xxxx
xxxx
xxxx(
Add the opposite of the polynomial being subtracted.
Group like terms.
Combine like terms.
Subtracting Polynomials
Blitzer, Introductory Algebra, 5e – Slide #12 Section 6.1
EXAMPLEEXAMPLE
Blitzer, Introductory Algebra, 5e – Slide #13 Section 6.1
Subtracting Polynomials
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Subtract . 8653765 324324 xyyxyxyyxyx
xyyxyxyyxyx 8653765 324324
xyyxyxyyxyx 8653765 324324 Change subtraction to addition and change the sign of every term of the polynomial in parentheses.
Rearrange terms
Combine like terms
xyyxyx
xyyyxyxyxyx
8 11 2
8675635
324
332424
Blitzer, Introductory Algebra, 5e – Slide #14 Section 6.1
Polynomial Functions
2524)( 23 xxxxf
is an example of a polynomial function. In a polynomial function, the expression that defines the function is a polynomial.
How do you evaluate a polynomial function? Use Substitution.
Blitzer, Introductory Algebra, 5e – Slide #15 Section 6.1
Graphs of Polynomial Functions
Polynomial functions of degree 2 or higher have graphs that are smooth and continuous.
By smooth, we mean that the graph contains only rounded corners with no sharp corners.
By continuous, we mean that the graph has no breaks and can be drawn without lifting the pencil from the page.
Blitzer, Introductory Algebra, 5e – Slide #16 Section 6.1
Graphs of Polynomials
EXAMPLEEXAMPLE
The graph below does not represent a polynomial function. Although it has a couple of smooth, rounded corners, it also has a sharp corner and a break in the graph. Either one of these last two features disqualifies it from being a polynomial function.
Smooth rounded curve
Smooth rounded curve
Discontinuous break
Sharp Corner