MTH 070Elementary Algebra

Chapter 5 – Exponents, Polynomials and Applications

Section 5.3 – Introduction to Polynomials and Polynomial Functions

Copyright © 2010 by Ron Wallace, all rights reserved.

Vocabulary Term

A number: 17 A variable: x A product: -2x3

Positive exponents only! No addition, subtraction, or division!

Coefficient of a Term The “largest” constant factor of a term.

That is, the number part of the term What about x, -x, x5, etc.?

Degree of a Term The number of variable factors of the term.

Vocabulary Polynomial

A term or sum of terms. Note: Subtraction is considered adding the opposite.

Convention: Put terms in order by degree.

Leading Term Term of a polynomial with highest degree.

Leading Coefficient Coefficient of the leading term.

Degree of the Polynomial Degree of the leading term.

Vocabulary

Monomial A polynomial with one term. 5x3

Binomial A polynomial with two terms. 3x2 – 5

Trinomial A polynomial with three terms. x2 – 4x + 3

Polynomial Expressions

For the above polynomial, determine … The number of terms: The degree of the second term: The degree of the polynomial: The leading coefficient: The coefficient of the second term: The coefficient of the linear term: Is this polynomial a monomial, binomial or

trinomial?

4 3 22 5 8 3x x x x

Evaluating a Polynomial

Given a polynomial and a value for its variable … substitute the value for the variable and do the arithmetic.

Example 1: Determine the value of x2 – 4x + 3 when x = 2

Example 2: Determine the value of x2 – 4x + 3 when x = –2

A Little Trick for Evaluating Polynomials

Determine the value of x2 – 4x + 3 when x = –2

2 1 4 3

A Little Trick for Evaluating Polynomials

Determine the value of x2 – 4x + 3 when x = –2

2 1 4 3

1

A Little Trick for Evaluating Polynomials

Determine the value of x2 – 4x + 3 when x = –2

2 1 4 3

2

1

A Little Trick for Evaluating Polynomials

Determine the value of x2 – 4x + 3 when x = –2

2 1 4 3

2

1 6

A Little Trick for Evaluating Polynomials

Determine the value of x2 – 4x + 3 when x = –2

2 1 4 3

2 12

1 6

A Little Trick for Evaluating Polynomials

Determine the value of x2 – 4x + 3 when x = –2

2 1 4 3

2 12

1 6 15

A Little Trick for Evaluating Polynomials

Determine the value of 5x3 - 7x2 - 2 when x = 3

Functions … a review from 3.6 Function … a named expression that gives

only one result for each value of the variable.

Notation: f(x) = an-expression-using-x Read as “f of x equals …” Doesn’t have to be f … g(x); h(x); p(x) … Doesn’t have to be x … f(a); g(m); d(t) …

Evaluating a Function f(3) means replace the variable in the expression

with 3 and do the arithmetic.

Polynomial Functions

A polynomial function is a function where the expression is a polynomial.

Example: P(x) = 2x2 – 4x + 3

Linear Function Polynomial function of degree 1

Quadratic Function Polynomial function of degree 2

Cubic Function Polynomial function of degree 3

Polynomial Functions

If P(x) = 2x2 + 4x + 3, find P(0) & P(1) & P(–5)

Adding Polynomials

“Combine Like Terms” i.e. Terms with the same variables can be

combined by adding their coefficients.

Order of terms in the answer? Descending order by degree Ascending order by degree Match the problem!

Adding Polynomials

Example …

4 2 3 22 5 3 7 4 6x x x x x

Review: Subtracting Signed Numbers

“Add the Opposite” a – b = a + (–b)

Opposite? The number the same distance from zero on

the other side of zero. –(5) = –5 –(–5) = 5 Essentially, multiplication by –1

Opposite of a Polynomial

If p(x) is a polynomial, then its opposite is … –p(x) = (-1)p(x)

Example: –(3x – 4) = (–1)(3x – 4) = –3x + 4

That is: Change the sign of every term.

Subtracting Polynomials

If p(x) and q(x) are polynomials, then

p(x) – q(x) = p(x) + (–q(x))

i.e. Add the opposite of the polynomial that follows the subtraction sign!

Subtracting Polynomials

Example …

4 2 3 22 5 3 7 4 6x x x x x