8-3 and 8-4 Multiplying Binomials

Quick Review

You can use algebra tiles, tables, or the Distributive Property to multiply polynomials. The FOIL method (First, Outer, Inner, Last) can be used to multiply two binomials. You can also use rules to multiply special case binomials.

Example

What is the simplified form of ( 4 x + 3 ) ( 3 x + 2 ) ? open 4 x plus 3 close open 3 x plus 2 close question mark

Use FOIL to multiply the binomials. Find the product of the first terms, the outer terms, the inner terms, and the last terms. Then add.

( 4 x + 3 ) ( 3 x + 2 ) = ( 4 x ) ( 3 x ) + ( 4 x ) ( 2 ) + ( 3 ) ( 3 x ) + ( 3 ) ( 2 )   = 12 x 2 + 8 x + 9 x + 6   = 12 x 2 + 17 x + 6 table with 3 rows and 2 columns , row1 column 1 , open 4 x plus 3 close open 3 x plus 2 close , column 2 equals open 4 x close open 3 x close plus open 4 x close open 2 close plus open 3 close open 3 x close plus open 3 close open 2 close , row2 column 1 , , column 2 equals , 12 x squared , plus 8 x plus 9 x plus 6 , row3 column 1 , , column 2 equals , 12 x squared , plus 17 x plus 6 , end table

Exercises

Simplify each product. Write in standard form.

29. ( w + 1 ) ( w + 12 ) open w plus 1 close open w plus 12 close
30. ( 2 s − 3 ) ( 5 s + 4 ) open 2 s minus 3 close open 5 s plus 4 close
31. ( 3 r − 2 ) 2 open 3 r minus 2 close squared
32. ( 6 g + 7 ) ( g − 8 ) open 6 g plus 7 close open g minus 8 close
33. ( 7 q + 2 ) ( 3 q + 8 ) open 7 q plus 2 close open 3 q plus 8 close
34. ( 4 n 3 + 5 ) ( 3 n + 5 ) open , 4 n cubed , plus 5 close open 3 n plus 5 close
35. ( t + 9 ) ( t − 3 ) open t plus 9 close open t minus 3 close
36. ( 6 c + 5 ) 2 open 6 c plus 5 close squared
37. ( 7 h − 3 ) ( 7 h + 3 ) open 7 h minus 3 close open 7 h plus 3 close
38. ( y − 6 ) ( 3 y + 7 ) open y minus 6 close open 3 y plus 7 close
39. ( 4 a − 7 ) ( 8 a + 3 ) open 4 eh minus 7 close open 8 eh plus 3 close
40. ( 4 b − 3 ) ( 4 b + 3 ) open 4 b minus 3 close open 4 b plus 3 close
41. Geometry A rectangle has dimensions 3x + 5 and x + 7. Write an expression for the area of the rectangle as a product and as a polynomial in standard form.

8-5 and 8-6 Factoring Quadratic Trinomials

Quick Review

You can write some quadratic trinomials as the product of two binomial factors. When you factor a polynomial, be sure to factor out the GCF first.

Example

What is the factored form of x 2 + 7 x + 12 ? x squared , plus 7 x plus 12 question mark

List the pairs of factors of 12. Identify the pair with a sum of 7.

x 2 + 7 x + 12 = ( x + 3 ) ( x + 4 ) x squared , plus 7 x plus 12 equals open x plus 3 close open x plus 4 close

Exercises

Factor each expression.

42. g 2 − 5 g − 14 g squared , minus 5 g minus 14
43. 2 n 2 + 3 n − 2 2 n squared , plus 3 n minus 2
44. 6 k 2 − 10 k ℓ + 4 ℓ 2 6 k squared , minus 10 k script l plus 4 , script l squared
45. p 2 + 8 p + 12 p squared , plus 8 p plus 12
46. r 2 + 6 r − 40 r squared , plus 6 r minus 40
47. 6 m 2 + 25 m n + 11 n 2 6 m squared , plus 25 m n plus , 11 n squared
48. t 2 − 13 t − 30 t squared , minus 13 t minus 30
49. 2 g 2 − 35 g + 17 2 g squared , minus 35 g plus 17
50. 3 x 2 + 3 x − 6 3 x squared , plus 3 x minus 6
51. d 2 − 18 d + 45 d squared , minus 18 d plus 45
52. w 2 − 15 w − 54 w squared , minus 15 w minus 54
53. 21 z 2 − 70 z + 49 21 z squared , minus 70 z plus 49
54. − 2 h 2 + 4 h + 70 negative , 2 h squared , plus 4 h plus 70
55. x 2 + 21 x + 38 x squared , plus 21 x plus 38
56. 10 v 2 + 11 v − 8 10 v squared , plus 11 v minus 8
57. 5 g 2 + 15 g + 10 5 g squared , plus 15 g plus 10
58. Reasoning Can you factor the expression 2 x 2 + 15 x + 9 ? 2 x squared , plus 15 x plus 9 question mark  Explain why or why not.

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