multiplyingpolynomials-130212164256-phpapp02.pptx

keklsnklsk

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

2. 7- 7 Multiplying Polynomials Warm Up Evaluate. Holt Algebra 1 1. 3 2 3. 10 2 Simplify. 4. 2 3  2 4 6. (5 3 ) 2 16 = 9 = 100 2 . 2 4 = 2 7 = 5 6 5. y 5  y 4 7. ( x 2 ) 4 8. –4( x – 7) = –4 x + 28 = y 9 = x 8

3. 7- 7 Multiplying Polynomials Holt Algebra 1 Multiply polynomials. Objective

4. 7- 7 Multiplying Polynomials Holt Algebra 1 To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.

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

6. 7- 7 Multiplying Polynomials When multiplying powers with the same base, keep the base and add the exponents. Holt Algebra 1 x 2 x 3 = x 2+3 = x 5 Remember!

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

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

9. 7- 7 Multiplying Polynomials Example 2A: Multiplying a Polynomial by a Monomial Multiply. 4(3 x 2 + 4 x – 8) 4 (3 x 2 + 4 x – 8) (4) 3 x 2 + (4) 4 x – (4) 8 Holt Algebra 1 12 x 2 + 16 x – 32 Distribute 4. Multiply.

10. 7- 7 Multiplying Polynomials Example 2B: Multiplying a Polynomial by a Monomial Multiply. 6 pq (2 p – q ) (6 pq ) (2 p – q ) (6 pq ) 2 p + (6 pq ) (– q ) Holt Algebra 1 (6  2)( p  p )( q ) + (–1)(6)( p )( q  q ) 12 p 2 q – 6 pq 2 Multiply. Distribute 6pq. Group like bases together.

11. Multiply. Holt Algebra 1 2 7- 7 Multiplying Polynomials Example 2C: Multiplying a Polynomial by a Monomial Multiply. 1 x 2 y  6 xy  8 x 2 y 2  2 1 x 2 y  6 xy  8 x 2 y 2  x y     8 x y  2  2 6 xy  x y 2 2 2    1   1    2  2 x • x     1  2 6  y • y  2 Group like bases together. x • x 2  y • y 2  8            2  1 3x 3 y 2 + 4x 4 y 3 Distribute 1 x 2 y . 2

12. 7- 7 Multiplying Polynomials Check It Out! Example 2 Multiply. a. 2(4 x 2 + x + 3) 2 (4 x 2 + x + 3) 2 (4 x 2 ) + 2 ( x ) + 2 (3) 8 x 2 + 2 x + 6 Distribute 2. Holt Algebra 1 Multiply.

13. 7- 7 Multiplying Polynomials Check It Out! Example 2 Multiply. b. 3 ab (5 a 2 + b ) 3 ab (5 a 2 + b ) (3 ab ) (5 a 2 ) + ( 3ab ) ( b ) Holt Algebra 1 (3  5)( a  a 2 )( b ) + (3)( a )( b  b ) 15 a 3 b + 3 ab 2 Distribute 3ab. Group like bases together. Multiply.

14. 7- 7 Multiplying Polynomials Check It Out! Example 2 Multiply. c. 5 r 2 s 2 ( r – 3 s ) 5 r 2 s 2 ( r – 3 s ) (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 ) Holt Algebra 1 5 r 3 s 2 – 15 r 2 s 3 Distribute 5r 2 s 2 . Group like bases together. Multiply.

15. 7- 7 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 x and 3. Distribute x and 3 again. Multiply. Combine like terms. = x ( x + 2 ) + 3 ( x + 2) = x ( x ) + x (2) + 3 ( x ) + 3 (2) = x 2 + 2 x + 3 x + 6 = x 2 + 5 x + 6 Holt Algebra 1

16. 7- 7 Multiplying Polynomials 3  x = 3 x x  2 = 2 x O 2. Multiply the Outer terms. ( x + 3)( x + 2 ) I 3. Multiply the Inner terms. ( x + 3 )( x + 2) 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 O I L Another method for multiplying binomials is called the FOIL method. F 1. Multiply the First terms. ( x + 3)( x + 2) x  x = x 2 Holt Algebra 1

17. 7- 7 Multiplying Polynomials Holt Algebra 1 Example 3A: Multiplying Binomials Multiply. ( s + 4)( s – 2) ( s + 4) ( s – 2) s ( s – 2) + 4 ( s – 2) s ( s ) + s (–2) + 4 ( s ) + 4 (–2) s 2 – 2 s + 4 s – 8 s 2 + 2 s – 8 Distribute s and 4. Distribute s and 4 again. Multiply. Combine like terms.

18. 7- 7 Multiplying Polynomials Holt Algebra 1 ( x – 4) 2 ( x – 4)( x – 4) Example 3B: Multiplying Binomials Multiply. Write as a product of ( x  x ) + ( x  (–4)) + (–4  x ) + (–4  (–4)) x 2 – 4 x – 4 x + 8 Multiply. x 2 – 8 x + 8 two binomials. Use the FOIL method. Combine like terms.

19. 7- 7 Multiplying Polynomials Holt Algebra 1 Example 3C: Multiplying Binomials Multiply. (8 m 2 – n )( m 2 – 3 n ) 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. Use the FOIL method.

20. 7- 7 Multiplying Polynomials Holt Algebra 1 In the expression ( x + 5) 2 , the base is ( x + 5). ( x + 5) 2 = ( x + 5)( x + 5) Helpful Hint

21. 7- 7 Multiplying Polynomials Holt Algebra 1 Check It Out! Example 3a Multiply. ( a + 3)( a – 4) ( a + 3) ( a – 4) a ( a – 4)+ 3 ( a – 4) a ( a ) + a (–4) + 3 ( a ) + 3 (–4) a 2 – 4 a + 3 a – 12 a 2 – a – 12 Distribute a and 3. Distribute a and 3 again. Multiply. Combine like terms.

22. 7- 7 Multiplying Polynomials Holt Algebra 1 ( x – 3) 2 Check It Out! Example 3b Multiply. Write as a product of ( x – 3)( x – 3) ( x ● x ) + ( x  (–3)) + (–3  x ) + (–3)(–3) x 2 – 3 x – 3 x + 9 Multiply. x 2 – 6 x + 9 Combine like terms. two binomials. Use the FOIL method.

23. 7- 7 Multiplying Polynomials Holt Algebra 1 Check It Out! Example 3c Multiply. (2 a – b 2 )( a + 4 b 2 ) (2 a – b 2 )( a + 4 b 2 ) 2 a ( a ) + 2 a (4 b 2 ) – b 2 ( a ) + (– b 2 )(4 b 2 ) 2 a 2 + 8 ab 2 – ab 2 – 4 b 4 Multiply. 2 a 2 + 7 ab 2 – 4 b 4 Combine like terms. Use the FOIL method.

24. 7- 7 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 + 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 Algebra 1

25. 7- 7 Multiplying Polynomials 10 x 3 + 56 x 2 – 18 Holt Algebra 1 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 +10 x – 6 10 x 3 50 x 2 – 30 x 6 x 2 30 x – 18 5 x +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. 10 x 3 + 6 x 2 + 50 x 2 + 30 x – 30 x – 18

26. 7- 7 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  5 x + 3 6 x 2 + 30 x – 18 + 10 x 3 + 50 x 2 – 30 x 10 x 3 + 56 x 2 + x – 18 10 x 3 + 56 x 2 + – 18 Holt Algebra 1 Multiply each term in the top polynomial by 3. Multiply each term in the top polynomial by 5x, and align like terms. Combine like terms by adding vertically. Simplify.

27. 7- 7 Multiplying Polynomials Holt Algebra 1 Multiply. ( x – 5)( x 2 + 4 x – 6) ( x – 5 ) ( x 2 + 4 x – 6) x ( x 2 + 4 x – 6) – 5 ( x 2 + 4 x – 6) Example 4A: Multiplying Polynomials 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. Distribute x and –5. Distribute x and −5 again.

28. 7- 7 Multiplying Polynomials Example 4B: Multiplying Polynomials Multiply. (2 x – 5)(–4 x 2 – 10 x + 3) (2 x – 5)(–4 x 2 – 10 x + 3) –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 Holt Algebra 1 Multiply each term in the top polynomial by –5. Multiply each term in the top polynomial by 2x, and align like terms. Combine like terms by adding vertically.

29. 7- 7 Multiplying Polynomials Holt Algebra 1 Example 4C: Multiplying Polynomials Multiply. ( x + 3) 3 [( x + 3)( x + 3)] ( x + 3) [ x ( x +3) + 3( x+ 3)] ( x + 3) ( x 2 + 3 x + 3 x + 9)( x + 3) ( x 2 + 6 x + 9)( x + 3) Write as the product of three binomials. Use the FOIL method on the first two factors. Multiply. Combine like terms.

30. 7- 7 Multiplying Polynomials Holt Algebra 1 Example 4C: Multiplying Polynomials Multiply. ( x + 3) 3 ( x + 3) ( x 2 + 6 x + 9) x ( x 2 + 6 x + 9) + 3 ( x 2 + 6 x + 9) x ( x 2 ) + x (6x) + x (9) + 3 ( x 2 ) + 3 (6 x ) + 3 (9) x 3 + 6 x 2 + 9 x + 3 x 2 + 18 x + 27 x 3 + 9 x 2 + 27 x + 27 Use the Commutative Property of Multiplication. Distribute the x and 3. Distribute the x and 3 again. Combine like terms.

31. 7- 7 Multiplying Polynomials Holt Algebra 1 Example 4D: Multiplying Polynomials Multiply. (3 x + 1)( x 3 – 4 x 2 – 7) x 3 4 x 2 – 7 3 x 4 12 x 3 – 21 x x 3 4 x 2 – 7 3 x +1 3 x 4 + 12 x 3 + x 3 + 4 x 2 – 21 x – 7 3 x 4 + 13 x 3 + 4 x 2 – 21 x – 7 Combine like terms. Write the product of the monomials in each row and column. Add all terms inside the rectangle.

32. 7- 7 Multiplying Polynomials Holt Algebra 1 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

33. 7- 7 Multiplying Polynomials Holt Algebra 1 Check It Out! Example 4a Multiply. ( x + 3)( x 2 – 4 x + 6) ( x + 3 ) ( x 2 – 4 x + 6) x ( x 2 – 4 x + 6) + 3 ( x 2 – 4 x + 6) Distribute x and 3. Distribute x and 3 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.

34. 7- 7 Multiplying Polynomials (3 x + 2)( x 2 – 2 x + 5) Check It Out! Example 4b Multiply. (3 x + 2)( x 2 – 2 x + 5) Multiply each term in the x 2 – 2 x + 5  3 x + 2 2 x 2 – 4 x + 10 + 3 x 3 – 6 x 2 + 15 x top polynomial by 2. Multiply each term in the top polynomial by 3x, and align like terms. Combine like terms by adding vertically. 3 x 3 – 4 x 2 + 11 x + 10 Holt Algebra 1

35. 7- 7 Multiplying Polynomials Holt Algebra 1 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. Write a polynomial that represents the area of the base of the prism. Write the formula for the area of a rectangle. Substitute h – 3 for w and h + 4 for l. Multiply. Combine like terms. A = l  w A = l  w A = ( h + 4)( h – 3) A = h 2 + 4 h – 3 h – 12 A = h 2 + h – 12 The area is represented by h 2 + h – 12.

36. 7- 7 Multiplying Polynomials The area is 18 square feet. Holt Algebra 1 A = h 2 + h – 12 A = 5 2 + 5 – 12 A = 25 + 5 – 12 A = 18 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. A = h 2 + h – 12 Write the formula for the area the base of the prism. Substitute 5 for h. Simplify. Combine terms.

37. 7- 7 Multiplying Polynomials Holt Algebra 1 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 area of a rectangle. Substitute x – 4 for l and x for w. Multiply. A = l  w A = l  w A = x ( x – 4) A = x 2 – 4 x The area is represented by x 2 – 4 x .

38. 7- 7 Multiplying Polynomials Holt Algebra 1 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 = x 2 – 4 x A = x 2 – 4 x A = 6 2 – 4  6 A = 36 – 24 A = 12 Write the formula for the area of a rectangle whose length is 4 meters shorter than width . Substitute 6 for x. Simplify. Combine terms. The area is 12 square meters.

39. 7- 7 Multiplying Polynomials Holt Algebra 1 Lesson Seatwork : Part I Multiply. 1. (6 s 2 t 2 )(3 st ) 2. 4 xy 2 ( x + y ) 4 x 2 y 2 + 4 xy 3 3. ( x + 2)( x – 8) (2 x – 7)( x 2 + 3 x – 4) 2 x 3 – x 2 – 29 x + 28 6 mn ( m 2 + 10 mn – 2) 6 m 3 n + 60 m 2 n 2 – 12 mn 6. (2 x – 5 y )(3 x + y ) 18 s 3 t 3 x 2 – 6 x – 16 6 x 2 – 13 xy – 5 y 2

40. 7- 7 Multiplying Polynomials Lesson Quiz: Part II A triangle has a base that is 4cm longer than its height. Write a polynomial that represents the area of the triangle. 2 b. Find the area when the height is 8 cm . 48 cm 2 Holt Algebra 1 1 h 2 + 2h