5 2 Polynomials Objectives 1 Add and Subtract

5. 2 Polynomials Objectives: 1. Add and Subtract Polynomials 2. Multiply Polynomials

Vocabulary • Polynomial – a monomial or a sum of monomials such as 5 x⁴-3 y. • Terms – the monomials that make up a polynomial • Like terms – monomials with the same variable(s) with the same exponent on those variables. • Binomial – a polynomial with 2 terms • Trinomial – a polynomial with 3 terms. • Degree of a polynomial – the highest degree of the monomials in the polynomial For 5 x⁶y²+2 x⁴y³, the degree for each monomial is 8 and 7. The degree of the polynomial is 8 since it is higher.

Determine whether each expression is a polynomial. If it is a polynomial, state the degree. 1. -4 x⁵y²-6 y³z⁸ yes, degree: 11 2. not a polynomial • 7 x⁴-9 x+3 yes, degree: 4

Adding and Subtracting Polynomials • To add polynomials, add like-terms. • When adding or subtracting like-terms, add or subtract the coefficients, but not the exponents. • For subtraction, change signs of 2 nd polynomial to opposites then use addition. Examples: § (2 x²-5 x+3)+(-4 x²+x-9)= -2 x²-4 x-6 § (4 a²-8 a+2)-(7 a²+6 a-8)= (4 a²-8 a+2)+(-7 a²-6 a+8)= -3 a²-14 a+10

Multiplying a polynomial by a monomial • Use the distributive property. (multiply coefficients and add exponents on like-bases) • Example: § -3 x²y³(5 xy⁴-7 x²y⁵)= -15 x³y⁷+21 x⁴y⁸ § ⅔ab(-6 a-9 b+12 ab)= -4 a²b-6 ab²+8 a²b²

Multiplying a Binomial by a Binomial Use FOIL Method or Distributive Property Examples: (FOIL) o (x-8)(4 x+5) 4 x²+5 x-32 x-40 4 x²-27 x-40 o (a³-6)(a³+3) a⁶+3 a³-6 a³-18 a⁶-3 a³-18 Distributive Property: (x-8)(4 x+5)= x(4 x+5)-8(4 x+5)= 4 x²+5 x-32 x-40= 4 x²-27 x-40

Column Method and Box Method Column Method (3 x+6)(7 x-5) 3 x+6 x 7 x-5 -15 x-30 21 x²+42 x+ 0 21 x²+27 x-30 Box Method (helpful for multiplying trinomials) (x-5)(2 x+9) x 2 x 9 -5 2 x² -10 x 9 x -45 2 x²-x-45

Power of a Binomial • To take a binomial to a power, write that binomial down that number of times and use your method of choice for multiplying binomials. • Example: (2 x-5)² (2 x-5) 4 x²-10 x+25 4 x²-20 x+25

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