Name Class Date. Adding and Subtracting Polynomials
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1 8-1 Reteaching Adding and Subtracting Polynomials You can add and subtract polynomials by lining up like terms and then adding or subtracting each part separately. What is the simplified form of (3x 4x + 5) + (5x + x 8)? Write the problem vertically, lining up the like terms. 3x 4x + 5 Then add each pair of like terms. + 5x + x 8 Solve Add the x terms. Add the x terms. Add the constant terms. 3x + 5x = 8x 4x + x = x 5 + ( 8) = 3 3x 4x x + x 8 8x x 3 Add the sums. Check Check your solution using subtraction. 8x 5x = 3x x x = 4x 3 ( 8) = 5 Solution: (3x 4x + 5) + (5x + x 8) = 8x x 3 Simplify. 5b 3b 1. b 5b 3c 3c. 4c c 4d 3d 6 3. d 3 5d 3 3e 5e 4. e e 7 4 f 3 f 5 f 5. f 3 4 f 3 f 5g 3 g 3g 6. g 3 5g g 7. (3h + 5) + ( 5h 3) 8. (j + 4j 6) + (4j 3j 3) 9
2 8-1 Reteaching (continued) Adding and Subtracting Polynomials To subtract polynomials, follow the same steps as in addition. What is the simplified form of (6x 3 + 4x 3x) (x 3 + 3x 5x)? Write the problem vertically, lining up the like terms. 6x 3 + 4x 3x Then subtract each pair of like terms. (x 3 + 3x 5x) Solve Subtract the x 3 terms. Subtract the x terms. Subtract the x terms. 6x 3 x 3 = 4x 3 4x 3x = x 3x ( 5x) = x 6x 3 + 4x 3x (x 3 + 3x 5x) 4x 3 + x + x Add the differences. Check Check your solution using subtraction. 4x 3 + x 3 = 6x 3 x + 3x = 4x x + ( 5x) = 3x Solution: (6x 3 + 4x 3x) (x 3 + 3x 5x) = 4x 3 + x + x Simplify. 9. 4k + 5k 10. 5m + 4m 11. 7n + 4n + 9 3k + k (m + 3m) (4n + 3n + 5) 1. 5p + 6p q 3 + q + 7q 14. r 3 r + 5r (7p + 4p+ 8) (6q 3 + 4q 5q) (4r 3 + 5r + 3r) 15. (6s 5s) ( s + 3s) 16. (3w + 6w 5) (5w 4w + ) 10
3 8- Reteaching Multiplying and Factoring You can multiply a monomial and a trinomial by solving simpler problems. You can use the Distributive Property to make three simpler multiplication problems. What is the simplified form of 3x(x + 4x 1)? Use the Distributive Property to rewrite the problem as three separate multiplication problems. 3x(x + 4x 1) = (3x x ) + (3x 4x) + (3x ( 1)) Remember that when you multiply same-base terms containing exponents, you add the exponents. Solve 3x x = 6x 3 Multiply inside the first pair of parentheses. 3x 4x = 1x Multiply inside the second pair of parentheses. 3x ( 1) = 3x Multiply inside the third pair of parentheses. 6x 3 + 1x 3x Add the products. Check 6x 3 x = 3x Check your solution using division. 1x 4x = 3x 3x ( 1) = 3x Solution: 3x(x + 4x 1) = 6x 3 + 1x 3x Simplify each product. 1. 4x(x 7). 3y(3y + 4) 3. z (z 3) 4. 3a( 4a 6) 5. 6b(3b + b 4) 6. 3c (c 4c + 3) 7. d(4d + 3d ) 8. 5e ( 3e e 3) 9. 4f( 3f 3 + f + 6) 19
4 8- Reteaching (continued) Multiplying and Factoring To factor a polynomial, find the greatest common factor (GCF) of the coefficients and constants and also the GCF of the variables. What is the factored form of 8x 4 + 1x 16x? Solve Find the GCF of the coefficients. Use prime factorization. 8 = 1 = 3 16 = The GCF of the numbers is 4. Each term has a variable. Remember, x = x 1. The GCF is the least exponent. The GCF of the variables is x. The GCF is 4x. Combine the GCFs. Factor out the GCF of each term. 4( + 3 4) Factor the coefficients. 4x(x 3 + 3x 4) Insert the variables. Check 4x(x 3 + 3x 4) = 8x 4 + 1x 16x Check by multiplying. Solution: The factored form of 8x 4 + 1x 16x is 4x(x 3 + 3x 4). Find the GCF of the terms of each polynomial x 6x 11. 4y + 1y z z 9z Factor each polynomial a b 18b 15. 9c 3 + 1c 16. 5d 3 10d + 0d 17. 6e + 10e g 3 4g + 16g 0
5 8-3 Reteaching Multiplying Binomials You can multiply binomials by using the FOIL method. FOIL stands for First, Outer, Inner, and Last. What is the simplified form of (4x + 3)(x + 6)? Use the FOIL method to simplify the binomial. Solve 4x x = 8x Multiply the First terms. 4x 6 = 4x Multiply the Outer terms. 3 x = 6x Multiply the Inner terms. 3 6 = 18 Multiply the Last terms. 8x + 4x + 6x x + 30x + 18 Add the products. Add the like terms. Check Substitute any number for x. Try x =. If the two sides of the equation are equal the simplification may be correct. (4x+3) (x+6) 8x + 30x + 18 (4 +3) ( + 6) (8 ) + (30 ) + 18 (11)(10) = 110 Solution: The simplified form of (4x + 3)(x + 6) is 8x + 30x Simplify each product. 1. (a + 6)(a 3). (b 4)(b + 5) 3. (c + 3)(c + 7) 4. (d + 4)(3d ) 5. (4e 5)(3e + 3) 6. (3f )(f 4) 7. (5g + 3)(g 3) 8. (4h + 4)(h + 5) 9. (3j 5)(4j 3) 9
6 8-3 Reteaching (continued) Multiplying Binomials To multiply a trinomial by a binomial, use the same steps as you would to multiply a 3- digit number by a -digit number. Find the partial products for each term of the binomial and then add the like terms of the partial products. What is the simplified form of (x + 3x 4)(3x + )? Solve Start by arranging the polynomials vertically. Multiply each part of the trinomial by. x + 3x 4 x = 4x 3x + 3x = 6x 4x + 6x 8 4 = 8 Multiply each part of the trinomial by 3x. x + 3x 4 x 3x = 6x 3 3x + 4x + 6x 8 3x 3x = 9x 6x 3 + 9x 1x 4 3x = 1x Add the partial products. 4x + 6x 8 6x 3 + 9x 1x 6x x 6x 8 Check Substitute any number for x. Try x =. If the two sides of the equation are equal, the simplification may be correct. (x + 3x 4) (3x + ) 6x x 6x 8 ( )(6 + ) =80 Solution: The simplified form of (x + 3x 4)(3x + ) is 6x x 6x 8. Simplify each product. 10. (w + 3w 4)(w + 3) 11. (x 8x + 6)(3x 4) 1. (y + 4y 5)(4y + ) 13. (3z 6z + 4)(4z + 1) 30
7 8-4 Reteaching Multiplying Special Cases A binomial is squared when it is multiplied by itself. The square of a binomial is the square of the first term plus the twice the product of the two terms plus the square of the last term. This can be expressed as (a + b) = a + ab + b. What is the simplified form of (x + 5)? Use the rules for squaring a binomial. Solve x x = x Square the first term. (5 x) = 10x Multiply the product of the two terms by. 5 5 = 5 Square the last term. So, (x + 5) = x + 10x + 5. Check (x + 5) = (x + 5)(x + 5) Rewrite the binomials. x x = x Multiply the First addends. x 5 = 5x Multiply the Outer addends. 5 x = 5x Multiply the Inner addends. 5 5 = 5 Multiply the Last addends. x + 5x + 5x + 5 x + 10x + 5 Add the products. Combine the like terms. Solution: The simplified form of (x + 5) is x + 10x + 5. Simplify each expression. 1. (a + 7). (b 4) 3. (c + 3) 4. (3d 5) 5. (4e + 1) 6. (f 6) 7. (g 10) 8. (5h + 8) 9. (3j 3) 10. (k + 4) 11. (4m ) 1. (3n + 6) 39
8 8-4 Reteaching (continued) Multiplying Special Cases The product of the sum and the difference of the same two terms produces a pattern that can be expanded algebraically as (a + b)(a b) = a ab + ab b. The sum of the two ab- terms is 0. Therefore, (a + b)(a b) = a b. The product is the square of the first term minus the square of the last term. What is the simplified form of (x 3)(x + 3)? Use the rules for finding the product of the sum and the difference of the same two terms. Solve x x = 4x Square the first term. 3 3 = 9 Square the last term. Remember, the product is the difference of the two squares. The product is 4x 9. Check Multiply the binomials using the FOIL Method. x x = 4x Multiply the First addends. x 3 = 6x Multiply the Outer addends. 3 x = 6x Multiply the Inner addends. 3 3 = 9 Multiply the Last addends. 4x + 6x 6x 9 4x 9 Add the products. Combine the like terms. Solution: The simplified form of (x 3)(x + 3) is 4x 9. Simplify each product. 13. (p 4)(p + 4) 14. (q + 5)(q 5) 15. (3r + )(3r ) 16. (4s 6)(4s + 6) 17. (t 1)(t + 1) 18. (5u 3)(5u + 3) 19. (6v 4)(6v + 4) 0. (3w 8)(3w + 8) 1. (7x 9)(7x + 9) 40
9 Extra Practice Chapter 8 Lesson 8-1 Find the degree of each monomial. 1. 7r 3. 4n w a 5 b 6. 4x y 3 z 4 Write each polynomial in standard form. Then name each polynomial based on degree and number of terms. 7. 4x + x 1 8. n + 3n y 10. w + 3 w + 8w d d 4 + 3d Simplify. Write each answer in standard form. 13. (5x 3 + 3x 7x + 10) (3x 3 x + 4x 1) 14. (x + 3x ) + (4x 5x + ) 15. (4m 3 + 7m 4) + (m 3 6m + 8) 16. (8t + t + 10) (9t 9t 1) 17. ( 7c 3 + c 8c 11) (3c 3 + c + c 4) 18. (6v + 3v 9v 3 ) + (7v 4v 10v 3 ) 19. (s 4 s 3 5s + 3s) (5s 4 + s 3 7s s) 0. (9w 4w + 10) + (8w w) 1. The sides of a rectangle are 4t 1 and 5t + 9. Write an expression for the perimeter of the rectangle.. Three consecutive integers are n 1, n, and n + 1. Write an expression for the sum of the three integers. Find an expression for the perimeter of each figure. 3. A rectangle has side lengths 4k 3 and k A triangle has side lengths t, 4t 3, and 10 t. 5. A rhombus has two sides 3d 3 d long and two sides d + 5 long. Prentice Hall Algebra 1 Extra Practice 9
10 Extra Practice (continued) Chapter 8 Lesson 8- Simplify each product. 6. y(y + 1) 7. 4b(b + 3) 8. 9c(c 3c + 5) 9. 8m(4m 5) 30. 5k(k + 8k) 31. 5r (r + 4r ) 3. m (m 3 + m ) 33. 3x(x + 3x 1) 34. x(1 + x + x ) Find the GCF terms of each polynomial. Factor y 4 9y 36. t 6 + t 4 t 5 + t 37. 3m 6 + 9m c 4c 3 + 1c v 6 + v 5 10v n 3n 3 + n r + 0r r 4. 9x 6 + 5x 5 + 4x d 8 d d A rectangular roof has a length of 13g and a width of 4g + 7. Write an expression for the area of the roof. 45. A cylinder has a base area of 3w + 5 and a height of 4w. Find an expression for the volume. Lessons 8-3 and 8-4 Simplify each product. Write in standard form. 46. (5c + 3)( c + ) 47. (3t 1)(t + 1) 48. (w + )(w + w 1) 49. (3t + 5)(t + 1) 50. (n 3)(n + 4) 51. (b + 3)(b + 7) 5. (3x + 1) 53. (5t + 4) 54. (w 1)(w + w + 1) 55. (a + 4)(a 4) 56. (3y )(3y + ) 57. (w + )(w ) 58. Geometry A rectangle has dimensions 3x 1 and x + 5. Write an expression for the area of the rectangle as a product and in standard form. 59. Write an expression for the product of the two consecutive odd integers n 1 and n + 1. Prentice Hall Algebra 1 Extra Practice 30
11 Extra Practice (continued) Chapter A circular pool has a radius of 5p 3 m. Write an expression for area the pool. 61. An office building has a rectangular base with side lengths of 1y 7 and y + 4. Write an expression for the area of a floor in the office building. 6. Suppose you play a game with two number cubes. Let A represent rolling a number less than 4 and B represent rolling a number greater than 4. The probability of A is 1. The probability of B is a. Find A B 3 3 b. What is the probability that both cubes show a number less than 4? c. What is the probability that one cube shows a number less than 4 and the other cube shows a number greater than 4? 63. Suppose there are two squares with side lengths of 4x 3 and x + 4. Write an expression for the area of each square. Find the area of each square if x = 6 cm. 64. A rectangle has dimensions of x + 3 and 4x 3. Write an expression for the area of the rectangle. Then find the area of the rectangle if x = 3 ft. Prentice Hall Algebra 1 Extra Practice 31
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