Lesson Objective: I will be able to … Multiply polynomials Language Objective: I will be able to … Read, write, and listen about vocabulary, key concepts, and examples

Example 1: Multiplying Monomials Page 34 Example 1: Multiplying Monomials Multiply. A. (6y3)(3y5) (6y3)(3y5) Group factors with like bases together. (6 3)(y3 y5)  18y8 Multiply. B. (3mn2) (9m2n) (3mn2)(9m2n) Group factors with like bases together. (3 9)(m m2)(n2  n)  27m3n3 Multiply.

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

Group factors with like bases together. (3x3)(6x2) Your Turn 1 Page 34 Multiply. A. (3x3)(6x2) Group factors with like bases together. (3x3)(6x2) (3 6)(x3 x2)  Multiply. 18x5 B. (2r2t)(5t3) Group factors with like bases together. (2r2t)(5t3) (2 5)(r2)(t3 t)  10r2t4 Multiply.

Example 2: Multiplying a Polynomial by a Monomial Page 34 Multiply. 4(3x2 + 4x – 8) 4(3x2 + 4x – 8) Distribute 4. (4)3x2 +(4)4x – (4)8 Multiply. 12x2 + 16x – 32

Example 3: Multiplying a Polynomial by a Monomial Page 35 Multiply. 6pq(2p – q) (6pq)(2p – q) Distribute 6pq. (6pq)2p + (6pq)(–q) 12p2q – 6pq2 Multiply.

Your Turn 3 Multiply. 3ab(5a2 + b) 3ab(5a2 + b) Distribute 3ab. Page 35 Multiply. 3ab(5a2 + b) 3ab(5a2 + b) Distribute 3ab. (3ab)(5a2) + (3ab)(b) 15a3b + 3ab2 Multiply.

x2 3x 2x 6 x + 3 (x + 3)(x + 2) = x ● x = 3 ● x = x + 2 Another method for multiplying polynomials is called the Generic Rectangle. For example, the product of (x + 3)(x + 2) can be found by finding the area of a rectangle with a length of (x + 3) and a width of (x + 2). Page 33 x + 3 (x + 3)(x + 2) = x ● x = x2 3 ● x = 3x x + 2 = x2 + 3x + 2x + 6 = x2 + 5x + 6 2 ● x = 2x 2 ● 3 = 6

To multiply a binomial by a binomial, you can apply the Distributive Property more than once: (x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3. Distribute x and 3 again. = x(x + 2) + 3(x + 2) = x(x) + x(2) + 3(x) + 3(2) Multiply. = x2 + 2x + 3x + 6 Combine like terms. = x2 + 5x + 6

1. Multiply the First terms. (x + 3)(x + 2) x x = x2 Another method for multiplying binomials is called FOIL. Page 33 F 1. Multiply the First terms. (x + 3)(x + 2) x x = x2  O 2. Multiply the Outer terms. (x + 3)(x + 2) x 2 = 2x  I 3. Multiply the Inner terms. (x + 3)(x + 2) 3 x = 3x  L 4. Multiply the Last terms. (x + 3)(x + 2) 3 2 = 6  (x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6 F O I L

Example 4: Multiplying Binomials Page 35 Example 4: Multiplying Binomials Multiply. (8m2 – n)(m2 – 3n) Use the FOIL method. 8m2(m2) + 8m2(–3n) – n(m2) – n(–3n) 8m4 – 24m2n – m2n + 3n2 Multiply. 8m4 – 25m2n + 3n2 Combine like terms.

Example 5: Multiplying Binomials Page 35 Example 5: Multiplying Binomials Multiply. Write as a product of two binomials. (x – 4)2 (x – 4)(x – 4) Use the FOIL method. (x x) + (x (–4)) + (–4  x) + (–4  (–4))  x2 – 4x – 4x + 16 Multiply. x2 – 8x + 16 Combine like terms.

Your Turn 5 Multiply. (a + 3)(a – 4) (a + 3)(a – 4) Page 36 Multiply. (a + 3)(a – 4) (a + 3)(a – 4) a(a) + a(–4) + 3(a) + 3(–4) Distribute a and 3. a2 – 4a + 3a – 12 Multiply. a2 – a – 12 Combine like terms.

To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x2 + 10x – 6): (5x + 3)(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6) = 5x(2x2) + 5x(10x) + 5x(–6) + 3(2x2) + 3(10x) + 3(–6) = 10x3 + 50x2 – 30x + 6x2 + 30x – 18 = 10x3 + 56x2 – 18

You can also use a rectangle model to multiply polynomials with more than two terms. This is similar to finding the area of a rectangle with length (2x2 + 10x – 6) and width (5x + 3): 2x2 +10x –6 10x3 50x2 –30x 30x 6x2 –18 5x +3 Write the product of the monomials in each row and column: To find the product, add all of the terms inside the rectangle by combining like terms and simplifying if necessary. 10x3 + 6x2 + 50x2 + 30x – 30x – 18 10x3 + 56x2 – 18

Another method that can be used to multiply polynomials with more than two terms is the vertical method. This is similar to methods used to multiply whole numbers. Multiply each term in the top polynomial by 3. 2x2 + 10x – 6 5x + 3  Multiply each term in the top polynomial by 5x, and align like terms. 6x2 + 30x – 18 + 10x3 + 50x2 – 30x Combine like terms by adding vertically. 10x3 + 56x2 + 0x – 18 10x3 + 56x2 – 18 Simplify.

Example 6: Geometry Application Page 36 Example 6: Geometry Application The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height. A. Write a polynomial that represents the area of the base of the prism. A = l  w A = l w  Write the formula for the area of a rectangle. Substitute h – 3 for w and h + 4 for l. A = (h + 4)(h – 3) A = h2 + 4h – 3h – 12 Multiply. A = h2 + h – 12 Combine like terms. The area is represented by h2 + h – 12.

Example 6: Geometry Application Page 36 B. Find the area of the base when the height is 5 ft. A = h2 + h – 12 A = 52 + 5 – 12 Substitute 5 for h. A = 25 + 5 – 12 Simplify. A = 18 The area is 18 square feet.

Classwork Assignment #6 Holt 7-7 #2 – 24 even