Unit 8: Polynomials Chapter Test. Part 1: Identify each of the following as: Monomial, binomial, or trinomial. Then give the degree of each.
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1 Unit 8: Polynomials Chapter Test Part 1: Identify each of the following as: Monomial, binomial, or trinomial. Then give the degree of each. 1. 9x x 2 + 3x y -1 Part 2: Simplify each expression. 1. (7x 3 + 2x 2 1) + (8x 2 + 2x -9) 11. (2x 2 + 4x 2)(3x 2 7x +1) 2. (3x 2 + 8x -1) (2x 2-6x + 4) 12. (2x+4) 2 (x-5) x(5x + 1) + 3(x 2 2x + 7) 13. (3x y) 2 (x+3) 4. 8x 2 (2x 2 9x + 3) 5. 7(x 3 4x + 8) 3x(2x 2 + 2x 6) 6. (y-3)(y +8) 7. (s +4)(s+5) 8. (x -7) 2 9. (p 4)(p+4) 10. (9x- 3y) 2 Part 3: Write an equation for each problem, then solve. 1. A rectangle has a length of x 2 + 2x + 3 and a width of 5x 1. Write an expression that represents the perimeter of the rectangle. Write an expression that represents the area of the rectangle. 2. A triangle has sides: 3x + 2, 8x 4, and 4x + 5. The perimeter of the triangle is 48 units. Find the length of the longest side of the triangle.
2 Unit 8: Polynomials Chapter Test Answer Key Part 1: Identify each of the following as: Monomial, binomial, or trinomial. Then give the degree of each. (1 point each) 1. 9x Binomial (2 terms); Degree 2 (2 is the largest exponent) Monomial (1 term); Degree 0 (Constant term with no variable degree of 0) 3. 2x 2 + 3x + 1 Trinomial (3 terms); Degree 2 (2 is the largest exponent) 4. 9y -1- Binomial; Degree 1 (y is raised to the first power although the exponent is not written) Part 2: Simplify each expression. (2 points each) 1. (7x 3 + 2x 2 1) + (8x 2 + 2x -9) Step 1: Write vertically and add. 7x 3 + 2x 2 + 0x 1 + 8x 2 + 2x - 9 7x 3 +10x 2 + 2x 10 Final Answer: 7x x 2 + 2x (3x 2 + 8x -1) (2x 2-6x + 4) Step 1: Rewrite as an addition problem. 3. 2x(5x + 1) + 3(x 2 2x + 7) Step 1: Distribute the 2x & the 3 throughout the parenthesis. (10x 2 + 2x) + (3x 2 6x + 21) Step 2: Write vertically and add. 10x 2 + 2x x 2-6x x 2 4x + 21 Final Answer: 13x 2 4x + 21 (3x 2 + 8x 1) + (-2x 2 + 6x 4) Step 2: Write vertically and add. 3x 2 + 8x x 2 + 6x 4 x x - 5 Final Answer: x x x 2 (2x 2 9x + 3) Step 1: Distribute the 8x 2 throughout the parenthesis. 16x 4 72x x 2 Final Answer: 16x 4 72x x 2
3 5. 7(x 3 4x + 8) 3x(2x 2 + 2x 6) Step 1: Distribute the 7 throughout the 1 st parenthesis, and -3x throughout the 2 nd parenthesis. (By distributing a negative 3x, you will be able to add the 2 polynomials) (7x 3 28x + 56) + (-6x 3 6x x) Step 2: Write vertically and add. 7x 3 + 0x 2 28x x 3-6x x + 0 x 3 6x 2 10x + 56 Final Answer: x 3 6x 2 10x (y-3)(y +8) Step 1: Use FOIL to multiply the two binomials. y(y) + y(8) + (-3)(y) + (-3)(8) y 2 + 8y - 3y - 24 Step 2: Combine like terms y 2 + 5y 24 Final Answer: y 2 + 5y (s +4)(s+5) Step 1: Use FOIL to multiply the two binomials. s(s) + s(5) + (4)(s) + (4)(5) s 2 + 5s + 4s +20 Step 2: Combine like terms s 2 + 9s + 20 Final Answer: s 2 + 9s + 20
4 8. (x -7) 2 Since we are squaring a binomial, we can use our special rule: Square of a Difference. (a-b) 2 = a 2 2ab + b 2 (x-7) 2 = x 2 2(x)(7) (x-7) 2 = x 2 14x + 49 Final Answer: x 2 14x (p 4)(p+4) This problem utilizes our special rule: Difference of Two Squares (a+b)(a-b) = a 2 b 2 (p-4)(p+4) = p (p-4)(p+4) = p 2-16 Final Answer: p (9x- 3y) 2 Since we are squaring a binomial, we can use our special rule: Square of a Difference. (a-b) 2 = a 2 2ab + b 2 (9x-3y) 2 =(9x) 2 2(9x)(3y) + (3y) 2 (9x-3y) 2 = 81x 2 54xy + 9y 2 Final Answer: = 81x 2 54xy + 9y (2x 2 + 4x 2)(3x 2 7x +1) Step 1: Use the extended distributive property to multiply. 2x 2 (3x 2 ) +2x 2 (-7x) + 2x 2 (1) + 4x(3x 2 ) +4x(-7x) + 4x (1)+ (-2)(3x 2 ) +(-2)(-7x) + (-2) (1) 6x 4-14x 3 + 2x x 3-28x 2 + 4x - 6x x - 2 Step 2: Rewrite with like terms together. 6x 4 14x x 3 + 2x 2 28x 2 6x 2 + 4x + 14x 2 Step 3: Combine like terms. Final Answer: 6x 4-2x 3-32x x - 2
5 12. (2x+4) 2 (x-5) 2 Step 1: Use the square of a binomial rule to expand both binomials. (a+b) 2 = a 2 + 2ab+ b 2 (2x+4) 2 = (2x) 2 +2(2x)(4)+ 4 2 (2x+4) 2 = 4x x + 16 (a-b) 2 = a 2 2ab +b 2 (x-5) 2 = x 2 2(x)(5) (x-5) 2 = x 2 10x + 25 Step 2: Use the expanded forms to subtract. (2x+4) 2 (x-5) 2 (4x x + 16) (x 2 10x + 25) Step 3: Rewrite as an addition problem. (4x x + 16) + (-x x - 25) Step 4: Write vertically and add. 4x x x x x x 9 Final Answer: 3x x (3x y) 2 (x+3) Step 1: Expand the first binomial. (3x-y) 2 = (3x) 2 2(3x)(y) + y 2 (3x-y) 2 = 9x 2 6xy + y 2 Step 2: Take this product and multiply it by (x+3) Use the extended distributive prop. (x+3) (9x 2 6xy + y 2 ) x(9x 2 ) +x(-6xy) +x(y 2 ) + 3(9x 2 ) +3(-6xy) +3(y 2 ) 9x 3-6x 2 y + xy x 2-18xy + 3y 2 Final Answer: 9x 3-6x 2 y + xy x 2-18xy + 3y 2 (There are no like terms)
6 Part 3: Write an equation for each problem, then solve. (2 points each) 1. A rectangle has a length of x 2 + 2x + 3 and a width of 5x 1. Write an expression that represents the perimeter of the rectangle. P = 2l + 2W P = 2(x 2 + 2x + 3) + 2(5x-1) - Substitute each expression for length and width P = (2x 2 + 4x + 6) + (10x 2) (Distribute) P = 2x 2 + 4x + 10x Write like terms together P = 2x x Final Answer Write an expression that represents the area of the rectangle. A = lw A = (x 2 + 2x + 3) (5x-1) - Substitute each expression for length and width Or A = (5x-1) (x 2 + 2x +3) - Switch terms around so that it s easier to use the distributive prop. A = 5x(x 2 ) + 5x(2x) + 5x(3) + (-1)(x 2 ) + (-1)(2x) + (-1)(3) A = 5x x x x 2 2x 3 A = 5x 3 + 9x 2 +13x 3 (combine like terms) Final Answer: A = 5x 3 + 9x 2 +13x 3 2. A triangle has sides: 3x + 2, 8x 4, and 4x + 5. The perimeter of the triangle is 48 units. Find the length of the longest side of the triangle. P = S 1 + S 2 + S 3 48= 3x+2 + 8x-4 + 4x + 5 Substitute the perimeter and lengths of sides. 48 = 3x + 8x + 4x Rewrite with like terms together 48 = 15x + 3 Simplify like terms 48 3 = 15x Subtract 3 from both sides to begin solving for x. 45 = 15x Simplify: (48-3 = 45) 45/15 = 15x/15 Divide by 15 on both sides. 3 = x or x = 3 S 1 = 3x+2 S 2 = 8x-4 S 3 = 4x+5 S 1 = 3(3) + 2 S 2 = 8(3) 4 S 3 = 4(3) + 5 S 1 = 11 S 2 = 20 S 3 = 17 The length of the longest side is 20 units. This test is worth 34 points.
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