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Problems of the Day

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

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  1. 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

  2. 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.

  3. Algebra 1 ~ Chapter 8.6 Multiplying a Polynomial by a Monomial

  4. To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.

  5. (6 3)(y3y5)   (-3 9)(m m2)(n2 n)   Multiplication of Monomials REVIEW A. (6y3)(3y5) 18y8 B. (-3mn2) (9m2n) -27m3n3

  6. Remember! When multiplying powers with the same base, keep the base and add the exponents. x2x3= x2+3 = x5

  7. 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.

  8. Ex. 2 – Multiplying a Polynomial by a Monomial −6pq(2p – q) (−6pq)(2p – q) (−6pq)2p + (−6pq)(–q) −12p2q +6pq2

  9. 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

  10. Remember - When simplifying expressions with more than one operation, you must still follow the order of operations.

  11. 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!

  12. 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!

  13. 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

  14. 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!!!

  15. 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

  16. Assignment • Study Guide 8-6 (In-Class) • Skills Practice 8-6 (Homework)

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