Multiplying and Factoring Section 8-2

Goals Goal To multiply a monomial by a polynomial. To factor a monomial from a polynomial. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary None

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

Multiply. A. (6y 3 )(3y 5 ) (6y 3 )(3y 5 ) 18y 8 Group factors with like bases together. B. (3mn 2 ) (9m 2 n) (3mn 2 )(9m 2 n) 27m 3 n 3 Multiply. Group factors with like bases together. Multiply. (6 3)(y 3 y 5 ) (3 9)(m m 2 )(n 2 n) Multiplying Monomials

Multiply. Group factors with like bases together. Multiply. Multiplying Monomials (4s 2 t 2 )(st)(-12st 2 ) (4 -12)(s 2 s s)(t 2 t t 2 ) -48s 4 t 5

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

Multiply. a. (3x 3 )(6x 2 ) (3x 3 )(6x 2 ) (3 6)(x 3 x 2 ) 18x 5 Group factors with like bases together. Multiply. Group factors with like bases together. Multiply. b. (2r 2 t)(5t 3 ) (2r 2 t)(5t 3 ) (2 5)(r 2 )(t 3 t) 10r 2 t 4 Your Turn:

Multiply. Group factors with like bases together. Multiply. c. Your Turn: (3x 2 y)(2x 3 z 2 )(y 4 z 5 ) (3 2)(x 2 x 3 )(y y 4 )(z 2 z 5 ) 6x5y5z76x5y5z7

To multiply a polynomial by a monomial, use the Distributive Property. Multiplying Monomials and Polynomials

Multiply. 4(3x 2 + 4x – 8) (4)3x 2 +(4)4x – (4)8 12x x – 32 Distribute 4. Multiply. Example: Multiplying a Polynomial by a Monomial

6pq(2p – q) (6pq)(2p – q) Multiply. (6pq)2p + (6pq)(–q) (6  2)(p  p)(q) + (–1)(6)(p)(q  q) 12p 2 q – 6pq 2 Distribute 6pq. Group like bases together. Multiply. Example: Multiplying a Polynomial by a Monomial

Multiply. a. 2(4x 2 + x + 3) 2(4x 2 + x + 3) 2(4x 2 ) + 2(x) + 2(3) 8x 2 + 2x + 6 Distribute 2. Multiply. Your Turn:

Multiply. b. 3ab(5a 2 + b) 3ab(5a 2 + b) (3ab)(5a 2 ) + (3ab)(b) (3  5)(a  a 2 )(b) + (3)(a)(b  b) 15a 3 b + 3ab 2 Distribute 3ab. Group like bases together. Multiply. Your Turn:

Multiply. c. 5r 2 s 2 (r – 3s) 5r 2 s 2 (r – 3s) (5r 2 s 2 )(r) – (5r 2 s 2 )(3s) (5)(r 2  r)(s 2 ) – (5  3)(r 2 )(s 2  s) 5r 3 s 2 – 15r 2 s 3 Distribute 5r 2 s 2. Group like bases together. Multiply. Your Turn:

When multiplying a polynomial by a negative monomial, be sure to distribute the negative sign. Helpful Hint

Multiply. d. – 5y 3 (y 2 + 6y – 8) – 5y 3 (y 2 + 6y – 8) – 5y 5 – 30y y 3 Multiply each term in parentheses by – 5y 3. Your Turn:

Factoring Factoring a polynomial reverses the multiplication process (factoring is unmultiplying). When factoring a monomial from a polynomial, the first step is to find the greatest common factor (GCF) of the polynomial’s terms.

Factors that are shared by two or more whole numbers are called common factors. The greatest of these common factors is called the greatest common factor, or GCF. Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 32: 1, 2, 4, 8, 16, 32 Common factors: 1, 2, 4 The greatest of the common factors is 4. Greatest Common Factor

Find the GCF of each pair of numbers. 100 and 60 factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100 factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 The GCF of 100 and 60 is 20. List all the factors. Circle the GCF. List the factors. Example: GCF of Two Numbers

Find the GCF of each pair of numbers. 12 and 16 factors of 12: 1, 2, 3, 4, 6, 12 factors of 16: 1, 2, 4, 8, 16 The GCF of 12 and 16 is 4. List all the factors. Circle the GCF. List the factors. Your Turn:

You can also find the GCF of monomials that include variables. To find the GCF of monomials, write the prime factorization of each coefficient and write all powers of variables as products. Then find the product of the common factors. GCF of Monomials

Find the GCF of each pair of monomials. 15x 3 and 9x 2 15x 3 = 3  5  x  x  x 9x 2 = 3  3  x  x 3  x  x = 3x 2 Write the factorization of each coefficient and write powers as products. Align the common factors. Find the product of the common factors. The GCF of 3x 3 and 6x 2 is 3x 2. Example: GCF of a Monomial

Find the GCF of each pair of monomials. 8x 2 and 7y 3 8x 2 = 2  2  2  x  x 7y 3 = 7  y  y  y Write the factorization of each coefficient and write powers as products. Align the common factors. There are no common factors other than 1. The GCF 8x 2 and 7y is 1. Example: GCF of a Monomial

If two terms contain the same variable raised to different powers, the GCF will contain that variable raised to the lower power. Helpful Hint

Find the GCF of each pair of monomials. 18g 2 and 27g 3 18g 2 = 2  3  3  g  g 27g 3 = 3  3  3  g  g  g 3  3  g  g The GCF of 18g 2 and 27g 3 is 9g 2. Write the factorization of each coefficient and write powers as products. Align the common factors. Find the product of the common factors. Your Turn:

Find the GCF of each pair of monomials. 16a 6 and 9b 9b = 3  3  b 16a 6 = 2  2  2  2  a  a  a  a  a  a Write the factorization of each coefficient and write powers as products. Align the common factors. There are no common factors other than 1. The GCF of 16a 6 and 7b is 1. Your Turn:

Find the GCF of each pair of monomials. 8x and 7x 2 8x = 2  2  2  x 7v 2 = 7  x  x Write the prime factorization of each coefficient and write powers as products. Align the common factors. The GCF of 8x and 7x 2 is x. Your Turn:

Recall that the Distributive Property states that ab + ac =a(b + c). The Distributive Property allows you to “ factor ” out the GCF of the terms in a polynomial to write a factored form of the polynomial. A polynomial is in its factored form when it is written as a product of monomials and polynomials that cannot be factored further. The polynomial 2(3x – 4x) is not fully factored because the terms in the parentheses have a common factor of x. Factoring out a Monomial

Factor each polynomial. Check your answer. 2x 2 – 4 2x 2 = 2  x  x 4 = 2  2 2 Find the GCF. The GCF of 2x 2 and 4 is 2. Write terms as products using the GCF as a factor. 2x 2 – (2  2) 2(x 2 – 2) Check 2(x 2 – 2) 2x 2 – 4 Multiply to check your answer. The product is the original polynomial. Use the Distributive Property to factor out the GCF. Example: Factoring the GCF

Aligning common factors can help you find the greatest common factor of two or more terms. Writing Math

Factor each polynomial. Check your answer. 8x 3 – 4x 2 – 16x 2x 2 (4x) – x(4x) – 4(4x) 4x(2x 2 – x – 4) 8x 3 – 4x 2 – 16x 8x 3 = 2  2  2  x  x  x 4x 2 = 2  2  x  x 16x = 2  2  2  2  x 2  2  x = 4x Find the GCF. The GCF of 8x 3, 4x 2, and 16x is 4x. Write terms as products using the GCF as a factor. Use the Distributive Property to factor out the GCF. Multiply to check your answer. The product is the original polynomials. Check Example: Factoring the GCF

Factor each polynomial. – 14x – 12x 2 – 1(14x + 12x 2 ) Both coefficients are negative. Factor out –1. Find the GCF. The GCF of 14x and 12x 2 is 2x. – 1[7(2x) + 6x(2x)] – 1[2x(7 + 6x)] – 2x(7 + 6x) Write each term as a product using the GCF. Use the Distributive Property to factor out the GCF. 14x = 2  7  x 12x 2 = 2  2  3  x  x 2  x = 2x Example: Factoring the GCF

When you factor out –1 as the first step, be sure to include it in all the other steps as well. Caution!

Factor each polynomial. 3x 3 + 2x 2 – = 2  5 Find the GCF. There are no common factors other than 1. The polynomial cannot be factored further. 3x 3 + 2x 2 – 10 3x 3 = 3  x  x  x 2x 2 = 2  x  x Example: Factoring the GCF

Factor each polynomial. Check your answer. 5b + 9b 3 5b = 5  b 9b = 3  3  b  b  b b 5(b) + 9b 2 (b) b(5 + 9b 2 ) Check 5b + 9b 3 Find the GCF. The GCF of 5b and 9b 3 is b. Multiply to check your answer. The product is the original polynomial. Write terms as products using the GCF as a factor. Use the Distributive Property to factor out the GCF. Your Turn:

Factor each polynomial. 9d 2 – 8 2 Find the GCF. There are no common factors other than 1. The polynomial cannot be factored further. 9d 2 – 8 2 9d 2 = 3  3  d  d 8 2 = 2  2  2  2  2  2 Your Turn:

Factor each polynomial. – 18y 3 – 7y 2 – 1(18y 3 + 7y 2 ) Both coefficients are negative. Factor out –1. Find the GCF. The GCF of 18y 3 and 7y 2 is y 2. 18y 3 = 2  3  3  y  y  y 7y 2 = 7  y  y y  y = y 2 Write each term as a product using the GCF. Use the Distributive Property to factor out the GCF.. – 1[18y(y 2 ) + 7(y 2 )] – 1[y 2 (18y + 7)] – y 2 (18y + 7) Example: Factoring the GCF

Factor each polynomial. 8x 4 + 4x 3 – 2x 2 8x 4 = 2  2  2  x  x  x  x 4x 3 = 2  2  x  x  x 2x 2 = 2  x  x 2  x  x = 2x 2 4x 2 (2x 2 ) + 2x(2x 2 ) – 1(2x 2 ) 2x 2 (4x 2 + 2x – 1) Check 2x 2 (4x 2 + 2x – 1) 8x 4 + 4x 3 – 2x 2 The GCF of 8x 4, 4x 3 and –2x 2 is 2x 2. Multiply to check your answer. The product is the original polynomial. Write terms as products using the GCF as a factor. Use the Distributive Property to factor out the GCF. Find the GCF. Your Turn:

To write expressions for the length and width of a rectangle with area expressed by a polynomial, you need to write the polynomial as a product. You can write a polynomial as a product by factoring it.

The area of a court for the game squash is 9x 2 + 6x m 2. Factor this polynomial to find possible expressions for the dimensions of the squash court. A = 9x 2 + 6x = 3x(3x) + 2(3x) = 3x(3x + 2) Possible expressions for the dimensions of the squash court are 3x m and (3x + 2) m. The GCF of 9x 2 and 6x is 3x. Write each term as a product using the GCF as a factor. Use the Distributive Property to factor out the GCF. Example: Application

What if…? The area of the solar panel on another calculator is (2x 2 + 4x) cm 2. Factor this polynomial to find possible expressions for the dimensions of the solar panel. A = 2x 2 + 4x = x(2x) + 2(2x) = 2x(x + 2) The GCF of 2x 2 and 4x is 2x. Write each term as a product using the GCF as a factor. Use the Distributive Property to factor out the GCF. Possible expressions for the dimensions of the solar panel are 2x cm, and (x + 2) cm. Your Turn:

Assignment 8-2 Exercises Pg : #10 – 40 even