Polynomials 7 8 Multiplying Polynomials Warm Up Lesson

Polynomials 7 -8 Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Holt Mc. Dougal Algebra 1 Algebra 1 1 Holt Mc. Dougal

7 -8 Multiplying Polynomials Warm Up Evaluate. 1. 32 9 2 3. 10 Simplify. 100 4. 23 • 24 6. (53)2 27 2. 2 4 56 8. – 4(x – 7) Holt Mc. Dougal Algebra 1 – 4 x + 28 16 5. y 5 • y 4 y 9 7. (x 2)4 x 8

7 -8 Multiplying Polynomials Objective Multiply polynomials. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Example 1: Multiplying Monomials Multiply. A. (6 y 3)(3 y 5) (6 • 3)(y 3 • y 5) 18 y 8 B. (3 mn 2) (9 m 2 n) (3 mn 2)(9 m 2 n) (3 • 9)(m • m 2)(n 2 • n) 27 m 3 n 3 Holt Mc. Dougal Algebra 1 Group factors with like bases together. Multiply.

7 -8 Multiplying Polynomials Remember! When multiplying powers with the same base, keep the base and add the exponents. x 2 • x 3 = x 2+3 = x 5 Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Check It Out! Example 1 Multiply. a. (3 x 3)(6 x 2) (3 • 6)(x 3 • x 2) 18 x 5 b. (2 r 2 t)(5 t 3) (2 • 5)(r 2)(t 3 • t) 10 r 2 t 4 Holt Mc. Dougal Algebra 1 Group factors with like bases together. Multiply.

7 -8 Multiplying Polynomials Check It Out! Example 1 Continued Multiply. � 1 2 � c. � x y ÷ (12 x 3 z � 3 � 2 )( y z ) 4 5 4 z 5 � 1 2 � 3 2 x y x z 12 ÷ � � � 3 )(y � 2 � 1 12 � • ÷ x • x � � 3 )( y • y )(z ( ( 4 x 5 y 5 z 7 Holt Mc. Dougal Algebra 1 3 ) 4 Group factors with like bases together. 2 ) • z 5 Multiply.

7 -8 Multiplying Polynomials To multiply a polynomial by a monomial, use the Distributive Property. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Example 2 A: Multiplying a Polynomial by a Monomial Multiply. 4(3 x 2 + 4 x – 8) Distribute 4. (4)3 x 2 +(4)4 x – (4)8 Multiply. 12 x 2 + 16 x – 32 Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Example 2 B: Multiplying a Polynomial by a Monomial Multiply. 6 pq(2 p – q) (6 • (6 pq)(2 p – q) Distribute 6 pq. (6 pq)2 p + (6 pq)(–q) Group like bases together. 2)(p • p)(q) + (– 1)(6)(p)(q • q) 12 p 2 q – 6 pq 2 Holt Mc. Dougal Algebra 1 Multiply.

7 -8 Multiplying Polynomials Example 2 C: Multiplying a Polynomial by a Monomial Multiply. 1 2 2 x y (6 xy + 8 x y 2 1 2 2 2 xy x y 6 + 8 x y 2 ( ) 2 ) Distribute 1 2 xy. 2 � 1 2 � 2 2 � 1 2 � Group like bases � x y ÷ (6 xy ) + � x y ÷ 8 x y � � 2 together. � 2 � 1 � 2 2 � 1 y • y 2 � • 6 ÷ x • x ( y • y ) + � • 8÷ x • x � � 2 ( ( 3 x 3 y 2 + 4 x 4 y 3 Holt Mc. Dougal Algebra 1 ) ) ( )( Multiply. )

7 -8 Multiplying Polynomials Check It Out! Example 2 Multiply. a. 2(4 x 2 + x + 3) Distribute 2. 2(4 x 2) + 2(x) + 2(3) Multiply. 8 x 2 + 2 x + 6 Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Check It Out! Example 2 Multiply. b. 3 ab(5 a 2 + b) Distribute 3 ab. (3 ab)(5 a 2) + (3 ab)(b) (3 • 5)(a • a 2)(b) + (3)(a)(b • b) 15 a 3 b + 3 ab 2 Holt Mc. Dougal Algebra 1 Group like bases together. Multiply.

7 -8 Multiplying Polynomials Check It Out! Example 2 Multiply. c. 5 r 2 s 2(r – 3 s) Distribute 5 r 2 s 2. (5 r 2 s 2)(r) – (5 r 2 s 2)(3 s) (5)(r 2 • r)(s 2) – (5 • 3)(r 2)(s 2 • s) 5 r 3 s 2 – 15 r 2 s 3 Holt Mc. Dougal Algebra 1 Group like bases together. Multiply.

7 -8 Multiplying Polynomials 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 again. = x(x) + x(2) + 3(x) + 3(2) Multiply. = x 2 + 2 x + 3 x + 6 Combine like terms. = x 2 + 5 x + 6 Holt Mc. Dougal Algebra 1 Distribute.

7 -8 Multiplying Polynomials Another method for multiplying binomials is called the FOIL method. F 1. Multiply the First terms. (x + 3)(x + 2) x • x = x 2 O 2. Multiply the Outside terms. (x + 3)(x + 2) x • 2 = 2 x I 3. Multiply the Inside terms. (x + 3)(x + 2) 3 • x = 3 x L 4. Multiply the Last terms. (x + 3)(x + 2) 3 • 2 = 6 (x + 3)(x + 2) = x 2 + 2 x + 3 x + 6 = x 2 + 5 x + 6 F Holt Mc. Dougal Algebra 1 O I L

7 -8 Multiplying Polynomials Example 3 A: Multiplying Binomials Multiply. (s + 4)(s – 2) s(s – 2) + 4(s – 2) Distribute. s(s) + s(– 2) + 4(s) + 4(– 2) Distribute again. s 2 – 2 s + 4 s – 8 Multiply. s 2 + 2 s – 8 Combine like terms. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Example 3 B: Multiplying Binomials Multiply. (x – 4)2 (x – 4) Write as a product of two binomials. Use the FOIL method. (x • x) + (x • (– 4)) + (– 4 • x) + (– 4 • (– 4)) x 2 – 4 x + 16 Multiply. x 2 – 8 x + 16 Combine like terms. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Example 3 C: Multiplying Binomials Multiply. (8 m 2 – n)(m 2 – 3 n) Use the FOIL method. 8 m 2(m 2) + 8 m 2(– 3 n) – n(m 2) – n(– 3 n) 8 m 4 – 24 m 2 n – m 2 n + 3 n 2 Multiply. 8 m 4 – 25 m 2 n + 3 n 2 Combine like terms. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Helpful Hint In the expression (x + 5)2, the base is (x + 5)2 = (x + 5) Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Check It Out! Example 3 a Multiply. (a + 3)(a – 4) a(a – 4)+3(a – 4) Distribute. a(a) + a(– 4) + 3(a) + 3(– 4) Distribute again. a 2 – 4 a + 3 a – 12 Multiply. a 2 – a – 12 Combine like terms. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Check It Out! Example 3 b Multiply. (x – 3)2 (x – 3) Write as a product of two binomials. Use the FOIL method. (x ● x) + (x • (– 3)) + (– 3 • x)+ (– 3) x 2 – 3 x + 9 Multiply. x 2 – 6 x + 9 Combine like terms. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Check It Out! Example 3 c Multiply. (2 a – b 2)(a + 4 b 2) Use the FOIL method. 2 a(a) + 2 a(4 b 2) – b 2(a) + (–b 2)(4 b 2) 2 a 2 + 8 ab 2 – 4 b 4 Multiply. 2 a 2 + 7 ab 2 – 4 b 4 Combine like terms. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5 x + 3) by (2 x 2 + 10 x – 6): (5 x + 3)(2 x 2 + 10 x – 6) = 5 x(2 x 2 + 10 x – 6) + 3(2 x 2 + 10 x – 6) = 5 x(2 x 2) + 5 x(10 x) + 5 x(– 6) + 3(2 x 2) + 3(10 x) + 3(– 6) = 10 x 3 + 50 x 2 – 30 x + 6 x 2 + 30 x – 18 = 10 x 3 + 56 x 2 – 18 Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials 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 (2 x 2 + 10 x – 6) and width (5 x + 3): 2 x 2 5 x +3 10 x 3 6 x 2 +10 x – 6 50 x 2 – 30 x – 18 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. 10 x 3 + 6 x 2 + 50 x 2 + 30 x – 18 10 x 3 + 56 x 2 – 18 Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials 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. 2 x 2 + 10 x – 6 Multiply each term in the top polynomial by 3. Multiply each term in the top 5 x + 3 polynomial by 5 x, and align 6 x 2 + 30 x – 18 like terms. + 10 x 3 + 50 x 2 – 30 x 10 x 3 + 56 x 2 + 0 x – 18 Combine like terms by adding vertically. 10 x 3 + 56 x 2 + – 18 Simplify. × Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Example 4 A: Multiplying Polynomials Multiply. (x – 5)(x 2 + 4 x – 6) (x – 5 )(x 2 + 4 x – 6) Distribute x. x(x 2 + 4 x – 6) – 5(x 2 + 4 x – 6) Distribute x again. x(x 2) + x(4 x) + x(– 6) – 5(x 2) – 5(4 x) – 5(– 6) x 3 + 4 x 2 – 5 x 2 – 6 x – 20 x + 30 Simplify. x 3 – x 2 – 26 x + 30 Combine like terms. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Example 4 B: Multiplying Polynomials Multiply. (2 x – 5)(– 4 x 2 – 10 x + 3) Multiply each term in the top polynomial by – 5. – 4 x 2 – 10 x + 3 x 2 x – 5 20 x 2 + 50 x – 15 + – 8 x 3 – 20 x 2 + 6 x – 8 x 3 + 56 x – 15 Multiply each term in the top polynomial by 2 x, and align like terms. Holt Mc. Dougal Algebra 1 Combine like terms by adding vertically.

7 -8 Multiplying Polynomials Example 4 C: Multiplying Polynomials Multiply. (x + 3)3 [x · x + x(3) + 3(x) + (3)(3)] Write as the product of three binomials. [x(x+3) + 3(x+3)](x + 3) Use the FOIL method on the first two factors. (x 2 + 3 x + 9)(x + 3) Multiply. (x 2 + 6 x + 9)(x + 3) Combine like terms. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Example 4 C: Multiplying Polynomials Continued Multiply. (x + 3)3 (x + 3)(x 2 + 6 x + 9) Use the Commutative Property of Multiplication. x(x 2 + 6 x + 9) + 3(x 2 + 6 x + 9) Distribute. x(x 2) + x(6 x) + x(9) + 3(x 2) + 3(6 x) + 3(9) Distribute again. x 3 + 6 x 2 + 9 x + 3 x 2 + 18 x + 27 Combine like terms. x 3 + 9 x 2 + 27 x + 27 Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Example 4 D: Multiplying Polynomials Multiply. (3 x + 1)(x 3 + 4 x 2 – 7) – 7 x 3 – 4 x 2 3 x 3 x 4 – 12 x 3 – 21 x +1 x 3 – 4 x 2 – 7 Write the product of the monomials in each row and column. Add all terms inside the rectangle. 3 x 4 – 12 x 3 + x 3 – 4 x 2 – 21 x – 7 3 x 4 – 11 x 3 – 4 x 2 – 21 x – 7 Holt Mc. Dougal Algebra 1 Combine like terms.

7 -8 Multiplying Polynomials Helpful Hint A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4 A, there are 2 • 3, or 6 terms before simplifying. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Check It Out! Example 4 a Multiply. (x + 3)(x 2 – 4 x + 6) (x + 3 )(x 2 – 4 x + 6) Distribute. x(x 2 – 4 x + 6) + 3(x 2 – 4 x + 6) Distribute again. x(x 2) + x(– 4 x) + x(6) +3(x 2) +3(– 4 x) +3(6) x 3 – 4 x 2 + 3 x 2 +6 x – 12 x + 18 Simplify. x 3 – x 2 – 6 x + 18 Combine like terms. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Check It Out! Example 4 b Multiply. (3 x + 2)(x 2 – 2 x + 5) x 2 – 2 x + 5 × 3 x + 2 2 x 2 – 4 x + 10 + 3 x 3 – 6 x 2 + 15 x 3 x 3 – 4 x 2 + 11 x + 10 Holt Mc. Dougal Algebra 1 Multiply each term in the top polynomial by 2. Multiply each term in the top polynomial by 3 x, and align like terms. Combine like terms by adding vertically.

7 -8 Multiplying Polynomials 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 = h 2 + 4 h – 3 h – 12 Write the formula for the area of a rectangle. Substitute h – 3 for w and h + 4 for l. Multiply. A = h 2 + h – 12 Combine like terms. A = l • w • A= l w A = (h + 4)(h – 3) The area is represented by h 2 + h – 12. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Example 5: Application Continued 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. A = h 2 + h – 12 Write the formula for the area the base of the prism. A = 52 + 5 – 12 Substitute 5 for h. A = 25 + 5 – 12 Simplify. A = 18 Combine terms. The area is 18 square feet. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials 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. Write the formula for the A = l w • area of a rectangle. • A= l w A = x(x – 4) A = x 2 – 4 x Substitute x – 4 for l and x for w. Multiply. The area is represented by x 2 – 4 x. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials Check It Out! Example 5 Continued 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 = x 2 – 4 x A = 62 – 4 • 6 Write the formula for the area of a rectangle whose length is 4 meters shorter than width. Substitute 6 for x. A = 36 – 24 Simplify. A = 12 Combine terms. A = x 2 – 4 x The area is 12 square meters. Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials 7. 8 Homework: Part I Multiply. 1. (6 s 2 t 2)(3 st) 2. 4 xy 2(x + y) 3. (x + 2)(x – 8) 4. (2 x – 7)(x 2 + 3 x – 4) 5. 6 mn(m 2 + 10 mn – 2) 6. (2 x – 5 y)(3 x + y) Holt Mc. Dougal Algebra 1

7 -8 Multiplying Polynomials 7. 8 Homework: 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. b. Find the area when the height is 8 cm. Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Example 4: Problem-Solving Application Write a polynomial that represents the area of the yard around the pool shown below. Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Example 4 Continued 1 Understand the Problem The answer will be an expression that shows the area of the yard less the area of the pool. List important information: • The yard is a square with a side length of x + 5. • The pool has side lengths of x + 2 and x – 2. Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Example 4 Continued 2 Make a Plan The area of the yard is (x + 5)2. The area of the pool is (x + 2) (x – 2). You can subtract the area of the pool from the yard to find the area of the yard surrounding the pool. Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Example 4 Continued 3 Solve Step 1 Find the total area. (x +5)2 = x 2 + 2(x)(5) + 52 = x 2 + 10 x + 25 Use the rule for (a + b)2: a = x and b = 5. Step 2 Find the area of the pool. (x + 2)(x – 2) = x 2 – 2 x + 2 x – 4 = x 2 – 4 Holt Mc. Dougal Algebra 1 Use the rule for (a + b)(a – b): a = x and b = 2.

7 -9 Special Products of Binomials Example 4 Continued 3 Solve Step 3 Find the area of the yard around the pool. Area of yard a = total area – area of pool = x 2 + 10 x + 25 – (x 2 – 4) Identify like = + 10 x + 25 – +4 terms. = (x 2 – x 2) + 10 x + ( 25 + 4) Group like terms = 10 x + 29 together The area of the yard around the pool is 10 x + 29. x 2 Holt Mc. Dougal Algebra 1 x 2

7 -9 Special Products of Binomials Example 4 Continued 4 Look Back Suppose that x = 20. Then the total area in the back yard would be 252 or 625. The area of the pool would be 22 • 18 or 396. The area of the yard around the pool would be 625 – 396 = 229. According to the solution, the area of the yard around the pool is 10 x + 29. If x = 20, then 10 x +29 = 10(20) + 29 = 229. Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Remember! To subtract a polynomial, add the opposite of each term. Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Check It Out! Example 4 Write an expression that represents the area of the swimming pool. Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Check It Out! Example 4 Continued 1 Understand the Problem The answer will be an expression that shows the area of the two rectangles combined. List important information: • The upper rectangle has side lengths of 5 + x and 5 – x. • The lower rectangle is a square with side length of x. Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Check It Out! Example 4 Continued 2 Make a Plan The area of the upper rectangle is (5 + x)(5 – x). The area of the lower square is x 2. Added together they give the total area of the pool. Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Check It Out! Example 4 Continued 3 Solve Step 1 Find the area of the upper rectangle. (5 + x)(5 – x) = 25 – 5 x + 5 x – x 2 Use the rule for (a + b) (a – b): a = 5 and b = x. = –x 2 + 25 Step 2 Find the area of the lower square. = x • x = x 2 Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Check It Out! Example 4 Continued 3 Solve Step 3 Find the area of the pool. Area of pool = rectangle area + square area a = = –x 2 + 25 + x 2 = (x 2 – x 2) + 25 = 25 The area of the pool is 25. Holt Mc. Dougal Algebra 1 + x 2 Identify like terms. Group like terms together

7 -9 Special Products of Binomials Check It Out! Example 4 Continued 4 Look Back Suppose that x = 2. Then the area of the upper rectangle would be 21. The area of the lower square would be 4. The area of the pool would be 21 + 4 = 25. According to the solution, the area of the pool is 25. Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials 7. 9 Homework: Part I Multiply. 1. (x + 7)2 2. (x – 2)2 3. (5 x + 2 y)2 4. (2 x – 9 y)2 5. (4 x + 5 y)(4 x – 5 y) 6. (m 2 + 2 n)(m 2 – 2 n) Holt Mc. Dougal Algebra 1

7 -9 Special Products of Binomials 7. 9 Homework: Part II 7. Write a polynomial that represents the shaded area of the figure below. x+6 x– 7 x– 6 x– 7 Holt Mc. Dougal Algebra 1