Preview Warm Up California Standards Lesson Presentation

Warm Up Evaluate. 1. 32 3. 102 Simplify. 4. 23  24 6. (53)2 9 2. 24 16 100 27 5. y5  y4 y9 56 7. (x2)4 x8 8. –4(x – 7) –4x + 28

California Standards 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Student solve multistep problems, including word problems, by using these techniques.

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

Additional Example 1: Multiplying Monomials 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.

Additional Example 1C: Multiplying Monomials ( ) æ ç è - 2 1 12 4 t s ö ÷ ø Group factors with like bases together. ( ) g æ - ö ç è 2 1 12 4 t s ÷ ø 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) Check It Out! Example 1 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)  Multiply. 10r2t4

( ) ( ) Check It Out! Example 1 Multiply. æ 1 ö c. ç x2y ÷ (12x3z2)(y4z5) è 3 ø ( ) æ ç è 4 5 2 1 12 3 x z y ö ÷ ø Group factors with like bases together. ( ) g æ ç è 3 2 4 5 1 12 z x x y y ö ÷ ø Multiply. 7 5 4 x y z

To multiply a polynomial by a monomial, use the Distributive Property.

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

Additional Example 2B: Multiplying a Polynomial by a Monomial 6pq(2p – q) (6pq)(2p – q) Distribute 6pq. (6pq)2p + (6pq)(–q) Group like bases together. (6  2)(p  p)(q) + (–1)(6)(p)(q  q) 12p2q – 6pq2 Multiply.

Additional Example 2C: Multiplying a Polynomial by a Monomial 1 x2y (6xy + 8x2y2) 2 x y ( ) + 2 6 1 xy y x 8 Distribute . 2 1 x y x y x y ( ) æ ç è + 2 1 6 8 xy ö ÷ ø Group like bases together. x2 • x ( ) æ + ç è 1 • 6 2 y • y x2 • x2 y • y2 • 8 ö ÷ ø 3x3y2 + 4x4y3 Multiply.

Check It Out! Example 2 Multiply. a. 2(4x2 + x + 3) 2(4x2 + x + 3) Distribute 2. 2(4x2) + 2(x) + 2(3) Multiply. 8x2 + 2x + 6

Check It Out! Example 2 Multiply. b. 3ab(5a2 + b) 3ab(5a2 + b) Distribute 3ab. (3ab)(5a2) + (3ab)(b) Group like bases together. (3  5)(a  a2)(b) + (3)(a)(b  b) 15a3b + 3ab2 Multiply.

Check It Out! Example 2 Multiply. c. 5r2s2(r – 3s) 5r2s2(r – 3s) Distribute 5r2s2. (5r2s2)(r) – (5r2s2)(3s) (5)(r2  r)(s2) – (5  3)(r2)(s2  s) Group like bases together. 5r3s2 – 15r2s3 Multiply.

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. = x(x + 2) + 3(x + 2) Distribute x and 3 again. = x(x) + x(2) + 3(x) + 3(2) = x2 + 2x + 3x + 6 Multiply. = x2 + 5x + 6 Combine like terms.

Another method for multiplying binomials is called the FOIL method. 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

Additional Example 3A: Multiplying Binomials (s + 4)(s – 2) (s + 4)(s – 2) s(s – 2) + 4(s – 2) Distribute s and 4. s(s) + s(–2) + 4(s) + 4(–2) Distribute s and 4 again. s2 – 2s + 4s – 8 Multiply. s2 + 2s – 8 Combine like terms.

Additional Example 3B: Multiplying Binomials Write as a product of two binomials. (x – 4)(x – 4) x(x) + x(–4) – 4(x) – 4(–4) Use the FOIL method. x2 – 4x – 4x + 16 Multiply. x2 – 8x + 16 Combine like terms.

Additional Example 3C: Multiplying Binomials (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.

In the expression (x + 5)2, the base is (x + 5) In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5) Helpful Hint

Check It Out! Example 3a Multiply. (a + 3)(a – 4) (a + 3)(a – 4) a(a – 4)+ 3(a – 4) Distribute a and 3. Distribute a and 3 again. a(a) + a(–4) + 3(a) + 3(–4) a2 – 4a + 3a – 12 Multiply. a2 – a – 12 Combine like terms.

Check It Out! Example 3b Multiply. (x – 3)2 Write as a product of two binomials. (x – 3)(x – 3) x(x) + x(–3) – 3(x) – 3(–3) Use the FOIL method. x2 – 3x – 3x + 9 Multiply. x2 – 6x + 9 Combine like terms.

Check It Out! Example 3c Multiply. (2a – b2)(a + 4b2) Use the FOIL method. 2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2) 2a2 + 8ab2 – ab2 – 4b4 Multiply. 2a2 + 7ab2 – 4b4 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. 2x2 + 10x – 6 Multiply each term in the top polynomial by 3. 5x + 3  Multiply each term in the top polynomial by 5x, and align like terms. 6x2 + 30x – 18 + 10x3 + 50x2 – 30x 10x3 + 56x2 + 0x – 18 Combine like terms by adding vertically. 10x3 + 56x2 – 18 Simplify.

Additional Example 4A: Multiplying Polynomials (x – 5)(x2 + 4x – 6) (x – 5 )(x2 + 4x – 6) Distribute x and –5. x(x2 + 4x – 6) – 5(x2 + 4x – 6) Distribute x and −5 again. x(x2) + x(4x) + x(–6) – 5(x2) – 5(4x) – 5(–6) x3 + 4x2 – 6x – 5x2 – 20x + 30 Simplify. x3 – x2 – 26x + 30 Combine like terms.

Additional Example 4B: Multiplying Polynomials (2x – 5)(–4x2 – 10x + 3) Multiply each term in the top polynomial by –5. (2x – 5)(–4x2 – 10x + 3) Multiply each term in the top polynomial by 2x, and align like terms. –4x2 – 10x + 3 2x – 5 × 20x2 + 50x – 15 + –8x3 – 20x2 + 6x Combine like terms by adding vertically. –8x3 + 56x – 15

Additional Example 4C: Multiplying Polynomials [(x + 3)(x + 3)](x + 3) Write as the product of three binomials. Use the FOIL method on the first two factors. [x(x) + x(3) + 3(x) +3(3)](x + 3) (x2 + 3x + 3x + 9)(x + 3) Multiply. (x2 + 6x + 9)(x + 3) Combine like terms.

Additional Example 4C Continued Multiply. (x + 3)3 Use the Commutative Property of Multiplication. (x + 3)(x2 + 6x + 9) x(x2 + 6x + 9) + 3(x2 + 6x + 9) Distribute the x and 3. x(x2) + x(6x) + x(9) + 3(x2) + 3(6x) + 3(9) Distribute the x and 3 again. x3 + 6x2 + 9x + 3x2 + 18x + 27 Combine like terms. x3 + 9x2 + 27x + 27

Additional Example 4D: Multiplying Polynomials (3x + 1)(x3 + 4x2 – 7) Write the product of the monomials in each row and column. x3 4x2 –7 3x 3x4 12x3 –21x +1 4x2 x3 –7 Add all terms inside the rectangle. 3x4 + 12x3 + x3 + 4x2 – 21x – 7 3x4 + 13x3 + 4x2 – 21x – 7 Combine like terms.

A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3, or 6 terms before simplifying. Helpful Hint 

Check It Out! Example 4a Multiply. (x + 3)(x2 – 4x + 6) (x + 3)(x2 – 4x + 6) Distribute x and 3. x(x2 – 4x + 6) + 3(x2 – 4x + 6) Distribute x and 3 again. x(x2) + x(–4x) + x(6) + 3(x2) + 3(–4x) + 3(6) x3 – 4x2 + 6x + 3x2 – 12x + 18 Simplify. x3 – x2 – 6x + 18 Combine like terms.

Check It Out! Example 4b Multiply. (3x + 2)(x2 – 2x + 5) Multiply each term in the top polynomial by 2. (3x + 2)(x2 – 2x + 5) Multiply each term in the top polynomial by 3x, and align like terms. x2 – 2x + 5 3x + 2  2x2 – 4x + 10 + 3x3 – 6x2 + 15x Combine like terms by adding vertically. 3x3 – 4x2 + 11x + 10

Additional Example 5: 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 – 3h + 4h – 12 Multiply. A = h2 + h – 12 Combine like terms. The area is represented by h2 + h – 12.

Additional Example 5: 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. B. Find the area of the base when the height is 5 ft. Use the polynomial from part A. A = h2 + h – 12 A = 52 + 5 – 12 Substitute 5 for h. A = 25 + 5 – 12 Simplify. A = 18 Combine terms. The area is 18 square feet.

The length of a rectangle is 4 meters shorter than its width. Check It Out! Example 5 The length of a rectangle is 4 meters shorter than its width. a. Write a polynomial that represents the area of the rectangle. A = l w  A = l w Write the formula for the area of a rectangle. Substitute x – 4 for l and x for w. A = (x – 4)x A = x2 – 4x Multiply. The area is represented by x2 – 4x.

Check It Out! Example 5 The length of a rectangle is 4 meters shorter than its width. b. Find the area of a rectangle when the width is 6 meters. A = x2 – 4x Use the polynomial from part a. A = 62 – 4(6) Substitute 6 for x. A = 36 – 24 Simplify. A = 12 Combine terms. The area is 12 square meters.

Lesson Quiz: Part I Multiply. 1. (6s2t2)(3st) 2. 4xy2(x + y) 3. (x + 2)(x – 8) 4. (2x – 7)(x2 + 3x – 4) 5. 6mn(m2 + 10mn – 2) 6. (2x – 5y)(3x + y) 18s3t3 4x2y2 + 4xy3 x2 – 6x – 16 2x3 – x2 – 29x + 28 6m3n + 60m2n2 – 12mn 6x2 – 13xy – 5y2

Lesson Quiz: Part II 7. A triangle has a base that is 4 cm longer than its height. a. Write a polynomial that represents the area of the triangle. 1 2 h2 + 2h b. Find the area when the height is 8 cm. 48 cm2