Problems of the Day Simplify each expression. 1. 9m 2 – 8m + 7m 2 2. (10r 2 + 4s 2 ) – (5r 2 + 6s 2 ) 3. (pq + 7p) + (6pq – 10p – 5pq) 4. (17d 2 – 4) – (9d 2 – 5d + 8) 5r 2 – 2s 2 16m 2 – 8m 2pq – 3p 8d 2 +5d – (6.5ab + 14b) – (–2.5ab + 9b)9ab + 5b

Find the degree of each polynomials. Then name the polynomials based on # of terms. A.) 8j 9 + 5j B.) -9g 6 h 5 + 6g C.) 2m + 3mn – 8m 5 n The degree of the polynomial is 9. The degree of the polynomial is 11. The degree of the polynomial is 6. This polynomial has 2 terms, so it is a binomial. This polynomial has 3 terms, so it is a trinomial.

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.

A. (6y 3 )(3y 5 ) 18y 8 B. (-3mn 2 ) (9m 2 n) -27m 3 n 3 (6 3)(y 3 y 5 )   (-3 9)(m m 2 )(n 2  n)  Multiplication of Monomials REVIEW

When multiplying powers with the same base, keep the base and add the exponents. x 2  x 3 = x 2+3 = x 5 Remember!

Ex. 1 – Multiplying a Polynomial by a Monomial 4(3x 2 + 4x – 8) (4)3x 2 + (4)4x – (4)8 12x x – 32 This expression is completely simplified. There are no “like terms” to combine.

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

1 2 x2yx2y (6xy + 8x 2 y 2 ) x y   yx 8 2          1 2       3x 3 y 2 + 4x 4 y 3 Ex. 3 – Multiplying a Polynomial by a Monomial

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

Ex. 4 – Simplify the expression 3(x 2 + 2x – 1) + 4(2x 2 – x + 3) = 3x 2 + 6x – 3 + 8x 2 – 4x + 12 = (3x 2 + 8x 2 ) + (6x – 4x) + ( ) = 11x 2 + 2x + 9 Distribute THEN Combine Like Terms!

Ex. 5 – Simplify the expression 3(2n 2 – 4n – 15) + 6n(5n + 2) = 6n 2 – 12n – n n = (6n n 2 ) + (-12n + 12n) + ( ) = 36n 2 – 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 – x 18x – 14 = – x = -36 x = -2 Distributive Property. Combine Like Terms Solve the 2-step equation CHECK your solution!!!

Lesson Review Simplify each expression. 1. (6s 2 t 2 )(−3st) 2. 4xy 2 (x + y) 3. 6mn(m mn – 2) 4. d( − 2d + 4) + 15d 5. 3w(6w – 4) + 2(w 2 – 3w + 5) 6. x(x – 1) + 14 = x(x – 8) 4x 2 y 2 + 4xy 3 −18s 3 t 3 6m 3 n + 60m 2 n 2 – 12mn − 2d d 20w 2 – 18w + 10 x = − 2

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