Precalculus Essentials Fifth Edition Chapter P Prerequisites Fundamental

Precalculus Essentials Fifth Edition Chapter P Prerequisites: Fundamental Concepts of Algebra 1 Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

P. 4 Polynomials Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Objectives • Understand the vocabulary of polynomials. • Add and subtract polynomials. • Multiply polynomials. • Use FOIL in polynomial multiplication. • Use special products in polynomial multiplication. • Perform operations with polynomials in several variables. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Definition of a Polynomial in x A polynomial in x is an algebraic expression of the form where an, an− 1, an− 2, . . . , a 1 and a 0 are real numbers, an ≠ 0, and n is a nonnegative integer. The polynomial is of degree n, an is the leading coefficient, and a 0 is the constant term. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Polynomials When a polynomial is in standard form, the terms are written in the order of descending powers of the variable. Thus, the notation that we use to describe a polynomial in x is: Simplified polynomials with one, two, or three terms have special names: monomial (one term); binomial (two terms); trinomial (three terms). Simplified polynomials with four or more terms have no special names. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Adding and Subtracting Polynomials are added and subtracted by combining like terms. Like terms are terms that have exactly the same variable factors. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Adding and Subtracting Polynomials Perform the indicated operations and simplify: Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Multiplying Polynomials The product of two monomials is obtained by using properties of exponents. We use the distributive property to multiply a monomial and a polynomial that is not a monomial. To multiply two polynomials when neither is a monomial, we multiply each term of one polynomial by each term of the other polynomial. Then, we combine like terms. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Multiplying a Binomial and a Trinomial Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

The Product of Two Binomials: FOIL Any two binomials can be quickly multiplied by using the FOIL method: F represents the product of the first two terms in each binomial. O represents the product of the outside terms. I represents the product of the inside terms. L represents the product of the last, or second, terms in each binomial. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Using the FOIL Method Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Special Products There are several products that occur so frequently that it’s convenient to memorize the form, or pattern, of these formulas. If A and B represent real numbers, variables, or algebraic expressions, then: Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Finding the Product of the Sum and Difference of Two Terms Solution: We will use the special product formula Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Polynomials in Several Variables The constant, a, is the coefficient. The exponents, n and m, represent whole numbers. The degree of a polynomial in two variables is the highest degree of all its terms. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Subtracting Polynomials in Two Variables Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Multiplying Polynomials in Two Variables Solution: Each of the factors is a binomial, so we can apply the FOIL method for this multiplication. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved