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This guide explains how to simplify polynomial expressions and calculate their degree. It also provides examples of multiplying polynomials by monomials and solving equations involving polynomial expressions.
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Problems of the Day Simplify each expression. 1. 9m2 – 8m + 7m2 2. (10r2 + 4s2) – (5r2 + 6s2) 3. (pq + 7p) + (6pq – 10p – 5pq) 4. (17d2 – 4) – (9d2 – 5d + 8) 16m2 – 8m 5r2 – 2s2 2pq – 3p 8d2 +5d – 12 5. (6.5ab + 14b) – (–2.5ab + 9b) 9ab + 5b
Find the degree of each polynomials. Then name the polynomials based on # of terms. A.) 8j9 + 5j B.) -9g6h5 + 6g8 + 7 C.) 2m + 3mn – 8m5n This polynomial has 2 terms, so it is a binomial. The degree of the polynomial is 9. This polynomial has 3 terms, so it is a trinomial. The degree of the polynomial is 11. This polynomial has 3 terms, so it is a trinomial. The degree of the polynomial is 6.
Algebra 1 ~ Chapter 8.6 Multiplying a Polynomial by a Monomial
To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.
(6 3)(y3y5) (-3 9)(m m2)(n2 n) Multiplication of Monomials REVIEW A. (6y3)(3y5) 18y8 B. (-3mn2) (9m2n) -27m3n3
Remember! When multiplying powers with the same base, keep the base and add the exponents. x2x3= x2+3 = x5
Ex. 1 – Multiplying a Polynomial by a Monomial 4(3x2 + 4x – 8) 4(3x2 + 4x – 8) (4)3x2 + (4)4x – (4)8 12x2 + 16x – 32 This expression is completely simplified. There are no “like terms” to combine.
Ex. 2 – Multiplying a Polynomial by a Monomial −6pq(2p – q) (−6pq)(2p – q) (−6pq)2p + (−6pq)(–q) −12p2q +6pq2
1 ( ) 2 2 xy 2 x y 6 + x y 8 2 1 ö æ ö æ 1 ( ) ( ) 2 2 2 2 x y 6 xy + x y 8 x y ÷ ç ÷ ç 2 2 ø è ø è Ex. 3 – Multiplying a Polynomial by a Monomial 1 x2y (6xy + 8x2y2) 2 3x3y2 + 4x4y3
Remember - When simplifying expressions with more than one operation, you must still follow the order of operations.
Ex. 4 – Simplify the expression 3(x2 + 2x – 1) + 4(2x2 – x + 3) = 3x2 + 6x – 3 + 8x2 – 4x + 12 = (3x2 + 8x2) + (6x – 4x) + (-3 + 12) = 11x2 + 2x + 9 Distribute THEN Combine Like Terms!
Ex. 5 – Simplify the expression 3(2n2 – 4n – 15) + 6n(5n + 2) = 6n2 – 12n – 45 + 30n2 + 12n = (6n2 + 30n2) + (-12n + 12n) + (-45 + 0) = 36n2 – 45 You do not write 0n in your final answer!
Solving Equations with Polynomial Expressions Many equations contain polynomials that must be added, subtracted, and/or multiplied before the equation can be solved. For example, 2(3x – 2) = 10x 6x – 4 = 10x -4 = 4x x = -1
Ex. 6 – Solve the equation 2(4x – 7) = 5(– 2x – 9) – 5 8x – 14 = – 10x – 45 – 5 8x – 14 = – 10x – 50 +10x +10x 18x – 14 = – 50 +14 +14 18x = -36 x = -2 Distributive Property. Combine Like Terms Solve the 2-step equation CHECK your solution!!!
Lesson Review Simplify each expression. 1. (6s2t2)(−3st) 2. 4xy2(x + y) 3. 6mn(m2 + 10mn – 2) 4. d(−2d + 4) + 15d 5. 3w(6w – 4) + 2(w2 – 3w + 5) 6. x(x – 1) + 14 = x(x – 8) −18s3t3 4x2y2 + 4xy3 6m3n + 60m2n2 – 12mn −2d2 + 19d 20w2 – 18w + 10 x = − 2
Assignment • Study Guide 8-6 (In-Class) • Skills Practice 8-6 (Homework)