Transcript Chapter 5-1

Chapter 5
Polynomials
and Polynomial
Functions
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-1
1
Chapter Sections
5.1 – Addition and Subtraction of Polynomials
5.2 – Multiplication of Polynomials
5.3 – Division of Polynomials and Synthetic
Division
5.4 – Factoring a Monomial from a Polynomial
and Factoring by Grouping
5.5 – Factoring Trinomials
5.6 – Special Factoring Formulas
5.7-A General Review of Factoring
5.8- Polynomial Equations
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-2
2
§ 5.1
Addition and
Subtraction of
Polynomials
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-3
3
Find the Degree of a Polynomial
A polynomial is a finite sum of terms in
which all variables have whole number
exponents and no variable appears in a
denominator.



3x2 + 2x + 6 is a polynomial in one variable x
x2y – 7x + 3 is a polynomial in two variables x and y
x1/2 is not a polynomial because the variable does not have
a whole number exponent
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-4
4
Identifying Polynomials
The degree of a term of a polynomial in
one variable is the exponent on the
variable in that term.
Example:
5x6 (Sixth) 4x3 (Third) 7x (First) 9 (Zero)
The degree of a polynomial is the same as that
of its highest-degree term.
Example:
5x6 + 4x3 – 7x + 9 (Sixth)
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-5
5
Find the Degree of a Polynomial
The leading term of a polynomial is the term
of highest degree. The leading coefficient is
the coefficient of the leading term.
Example: 2x5 – 3x2 + 6x – 9
The degree of the polynomial is 5, the leading term is
2x5 and the leading coefficient is 2.
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Chapter 5-6
6
Identifying Polynomials
A polynomial is written in descending order
(or descending powers) of the variable
when the exponents on the variable decrease
from left to right.
Example:
5x6 + 4x3 – 7x + 9
A polynomial with one term is called a
monomial. A binomial is a two-termed
polynomial. A trinomial is a three-termed
polynomial.
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Chapter 5-7
7
Evaluate Polynomial Functions
A polynomial function is an expression used to
describe the function in a polynomial.
Example: For the polynomial function P(x) = 4x3 –
6x2 -2x + 9, find P(0).
P(0) = 4(0)3 – 6(0)2 -2(0) + 9
= 0–0–0+9
= 9
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Chapter 5-8
8
Understand Graphs of Polynomial Functions
These graphs have a positive leading coefficient,
and therefore, the function continues to increase
to the right of some value of x.
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Chapter 5-9
9
Understand Graphs of Polynomial Functions
These graphs have a negative leading coefficient,
and therefore, the function continues to decrease
to the right of some value of x.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 5-10
10
Adding Polynomials
To add polynomials, remove parentheses if
any are present. Then combine like terms.
Example:
(4 x  6 x  8)  (2 x  5 x  1)
2
2
 4 x 2  6 x  8  2 x 2  5x 1
 4x 2  2 x 2  6 x  5x  8 1
 6x  x  7
2
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Chapter 5-11
11
Subtracting Polynomials
1. Use the distributive property to remove
parentheses. (This will have the effect of
changing the sign of every term within the
parentheses of the polynomial being
subtracted.)
–(4x3 + 5x2 – 8) = – 4x3 – 5x2 + 8
2. Combine like terms.
Example:
(5x – 6) – (2x – 3) = 5x – 6 – 2x + 3 = 3x – 3
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Chapter 5-12
12