MTH 070 Elementary 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 Coefficient of a Term Degree of a 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 Leading Term Leading Coefficient 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 Binomial Trinomial A polynomial with one term. 5x3 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?

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

A Little Trick for Evaluating Polynomials 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

A Little Trick for Evaluating Polynomials 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

A Little Trick for Evaluating Polynomials Determine the value of x2 – 4x + 3 when x = –2

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” Order of terms in the answer? 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 …

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 …