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Factor Trinomials

Grade 9 · Algebra · Worksheet 3

  1. Aroha is designing a rectangular mosaic on a coordinate grid. The area of the mosaic, in square units, is given by the quadratic expression 14x² + 41x + 15. She partitions the rectangle into four smaller rectangles using the area model method, where the length and width of the whole rectangle are binomial expressions with integer coefficients. What is the factored form representing the dimensions of Aroha's mosaic? Answer: ______________
  2. Factor: 15x² + 34x + 15 Answer: ______________
  3. Olivia is designing a rectangular flower bed for a community garden project. The area of the flower bed is given by the quadratic expression 9x² + 21x + 10 square feet, where x represents the width in feet. The length and width of the flower bed are both binomial expressions with integer coefficients. Using the AC method, factor this quadratic expression to find the possible dimensions of the flower bed. Answer: ______________
  4. A rectangular garden has an area that can be expressed as 6x² + 19x + 10 square feet. If the length of the garden is represented by a binomial expression (ax + b) and the width by another binomial (cx + d), where all coefficients are positive integers, what is the factored form that represents the possible dimensions of the garden? Answer: ______________
  5. A rectangular garden has an area represented by the quadratic expression 2x² + 7x + 6 square meters. If the length and width are both binomial expressions with integer coefficients, what is the factored form that represents the possible dimensions of the garden? Answer: ______________
  6. A rectangular garden has an area represented by the expression 2x² + 7x + 6 square meters. If the length and width are both binomial expressions with integer coefficients, what is the factored form of this quadratic expression? Answer: ______________
  7. Factor: 8x² + 26x + 15 = ? Answer: ______________
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Answer Key & Explanations

Factor Trinomials · Grade 9 · Worksheet 3

  1. Aroha is designing a rectangular mosaic on a coordinate grid. The area of the mosaic, in square units, is given by the quadratic expression 14x² + 41x + 15. She partitions the rectangle into four smaller rectangles using the area model method, where the length and width of the whole rectangle are binomial expressions with integer coefficients. What is the factored form representing the dimensions of Aroha's mosaic? Answer: (2x+5)(7x+3) Solution: Identify a, b, and c in 14x² + 41x + 15. Here a = 14, b = 41, c = 15. Multiply a and c: 14 × 15 = 210.
    Full step-by-step solution

    Step 1: Identify a, b, and c in 14x² + 41x + 15. Here a = 14, b = 41, c = 15. Step 2: Multiply a and c: 14 × 15 = 210. Step 3: Find two numbers that multiply to 210 and add to 41. The numbers are 6 and 35 because 6 × 35 = 210 and 6 + 35 = 41. Step 4: Rewrite the middle term using these numbers: 14x² + 6x + 35x + 15. Step 5: Factor by grouping. Group the first two terms and the last two terms: (14x² + 6x) + (35x + 15). Step 6: Factor out the greatest common factor from each group: 2x(7x + 3) + 5(7x + 3). Step 7: Factor out the common binomial (7x + 3): (2x + 5)(7x + 3). The factored form is (2x+5)(7x+3).

  2. Factor: 15x² + 34x + 15 Answer: (5x + 3)(3x + 5) Solution: Identify a = 15, b = 34, c = 15. Multiply a and c: 15 × 15 = 225. Find two numbers that multiply to 225 and add to 34.
    Full step-by-step solution

    Step 1: Identify a = 15, b = 34, c = 15. Step 2: Multiply a and c: 15 × 15 = 225. Step 3: Find two numbers that multiply to 225 and add to 34. The numbers are 9 and 25 because 9 × 25 = 225 and 9 + 25 = 34. Step 4: Rewrite the middle term: 15x² + 9x + 25x + 15. Step 5: Factor by grouping: (15x² + 9x) + (25x + 15). Step 6: Factor out common factors: 3x(5x + 3) + 5(5x + 3). Step 7: Factor out the common binomial: (5x + 3)(3x + 5). The answer is (5x + 3)(3x + 5).

  3. Olivia is designing a rectangular flower bed for a community garden project. The area of the flower bed is given by the quadratic expression 9x² + 21x + 10 square feet, where x represents the width in feet. The length and width of the flower bed are both binomial expressions with integer coefficients. Using the AC method, factor this quadratic expression to find the possible dimensions of the flower bed. Answer: (3x + 2)(3x + 5) Solution: Identify coefficients: a = 9, b = 21, c = 10 Calculate a×c = 9×10 = 90 Find two numbers that multiply to 90 and add to 21: 15 and 6 (since 15×6 = 90 and 15+6 = 21) Rewrite the middle term: 9x² + 15x + 6x + 10 Group terms: (9x² + 15x) + (6x + 10) Factor each group: 3x(3x + 5) + 2(3x + 5) Factor…
    Full step-by-step solution

    Step 1: Identify coefficients: a = 9, b = 21, c = 10 Step 2: Calculate a×c = 9×10 = 90 Step 3: Find two numbers that multiply to 90 and add to 21: 15 and 6 (since 15×6 = 90 and 15+6 = 21) Step 4: Rewrite the middle term: 9x² + 15x + 6x + 10 Step 5: Group terms: (9x² + 15x) + (6x + 10) Step 6: Factor each group: 3x(3x + 5) + 2(3x + 5) Step 7: Factor out the common binomial: (3x + 2)(3x + 5) The factored form is (3x + 2)(3x + 5).

  4. A rectangular garden has an area that can be expressed as 6x² + 19x + 10 square feet. If the length of the garden is represented by a binomial expression (ax + b) and the width by another binomial (cx + d), where all coefficients are positive integers, what is the factored form that represents the possible dimensions of the garden? Answer: (2x + 5)(3x + 2) Solution: We are given the area of the rectangular garden as: 6x² + 19x + 10 square feet. We are told the length and width are binomials (ax + b) and (cx + d) with positive integer coefficients.
    Full step-by-step solution

    We are given the area of the rectangular garden as: 6x² + 19x + 10 square feet. We are told the length and width are binomials (ax + b) and (cx + d) with positive integer coefficients. --- **Step 1: Understand the factoring task** We need to factor the trinomial: 6x² + 19x + 10 into the form (ax + b)(cx + d) where a, b, c, d are positive integers. --- **Step 2: Multiply the general binomials** (ax + b)(cx + d) = ac x² + (ad + bc) x + bd So: ac = 6 ad + bc = 19 bd = 10 --- **Step 3: Find possible a, c values** Possible positive integer pairs (a, c) such that ac = 6: (1, 6), (2, 3), (3, 2), (6, 1) --- **Step 4: Find possible b, d values** Possible positive integer pairs (b, d) such that bd = 10: (1, 10), (2, 5), (5, 2), (10, 1) --- **Step 5: Check middle term condition** We need ad + bc = 19. Try a=2, c=3: Then ad + bc = 2d + 3b = 19 Possible b,d from bd=10: - b=2, d=5: 2*5 + 3*2 = 10 + 6 = 16 (no) - b=5, d=2: 2*2 + 3*5 = 4 + 15 = 19 (yes) So a=2, b=5, c=3, d=2 works. --- **Step 6: Write the factored form** (ax + b) = (2x + 5) (cx + d) = (3x + 2) Check: (2x + 5)(3x + 2) = 6x² + 4x + 15x + 10 = 6x² + 19x + 10 ✓ --- **Final Answer:** (2x + 5)(3x + 2)

  5. A rectangular garden has an area represented by the quadratic expression 2x² + 7x + 6 square meters. If the length and width are both binomial expressions with integer coefficients, what is the factored form that represents the possible dimensions of the garden? Answer: (2x+3)(x+2) Solution: We are given the quadratic expression: 2x² + 7x + 6. We want to factor it into two binomials with integer coefficients: (ax + b)(cx + d) = 2x² + 7x + 6.
    Full step-by-step solution

    We are given the quadratic expression: 2x² + 7x + 6. We want to factor it into two binomials with integer coefficients: (ax + b)(cx + d) = 2x² + 7x + 6. --- **Step 1: Multiply a and c** In the general form (ax + b)(cx + d), the product a*c must equal 2 (the coefficient of x²). Possible integer pairs for (a, c): (1, 2) or (2, 1). --- **Step 2: Multiply b and d** The product b*d must equal 6 (the constant term). Possible integer pairs for (b, d): (1, 6), (2, 3), (3, 2), (6, 1), and also negative possibilities, but here all signs are positive because the middle term is +7x and constant is +6. So b and d are both positive. --- **Step 3: Try possible factorizations** We test (1x + b)(2x + d) first. We need b*d = 6 and also 1*d + 2*b = 7 (because FOIL: Outer = 1*d*x, Inner = 2*b*x, sum = (d + 2b)x = 7x). Test b=1, d=6: d + 2b = 6 + 2 = 8 (no) Test b=2, d=3: d + 2b = 3 + 4 = 7 (yes!) Test b=3, d=2: d + 2b = 2 + 6 = 8 (no) Test b=6, d=1: d + 2b = 1 + 12 = 13 (no) So b=2, d=3 works. --- **Step 4: Write the factorization** (1x + 2)(2x + 3) = 2x² + 3x + 4x + 6 = 2x² + 7x + 6. Correct. We can also write it as (x + 2)(2x + 3) or (2x + 3)(x + 2). --- **Step 5: Interpret as dimensions** The problem says length and width are binomials with integer coefficients, so the factored form representing possible dimensions is: (2x + 3) and (x + 2) --- **Final answer:** (2x+3)(x+2)

  6. A rectangular garden has an area represented by the expression 2x² + 7x + 6 square meters. If the length and width are both binomial expressions with integer coefficients, what is the factored form of this quadratic expression? Answer: (2x+3)(x+2) Solution: We are given the quadratic expression: 2x² + 7x + 6 We want to factor it into two binomials with integer coefficients. a = 2, b = 7, c = 6 a × c = 2 × 6 = 12 b = 7 So we need two numbers whose product is 12 and sum is 7.
    Full step-by-step solution

    We are given the quadratic expression: 2x² + 7x + 6 We want to factor it into two binomials with integer coefficients. --- **Step 1: Understand the factoring method for a quadratic ax² + bx + c when a ≠ 1** We need two numbers that multiply to a × c and add to b. Here: a = 2, b = 7, c = 6 a × c = 2 × 6 = 12 b = 7 So we need two numbers whose product is 12 and sum is 7. --- **Step 2: Find the pair of numbers** List factor pairs of 12: 1 and 12 → sum = 13 (no) 2 and 6 → sum = 8 (no) 3 and 4 → sum = 7 (yes) So the numbers are 3 and 4. --- **Step 3: Rewrite the middle term using these numbers** 2x² + 7x + 6 = 2x² + 3x + 4x + 6 --- **Step 4: Factor by grouping** Group terms: (2x² + 3x) + (4x + 6) Factor out common factors from each group: From first group: x(2x + 3) From second group: 2(2x + 3) Now we have: x(2x + 3) + 2(2x + 3) --- **Step 5: Factor out the common binomial** Common factor is (2x + 3): (2x + 3)(x + 2) --- **Step 6: Check by expanding** (2x + 3)(x + 2) = 2x(x) + 2x(2) + 3(x) + 3(2) = 2x² + 4x + 3x + 6 = 2x² + 7x + 6 ✓ --- **Final answer:** (2x+3)(x+2)

  7. Factor: 8x² + 26x + 15 = ? Answer: (4x + 3)(2x + 5) Solution: Identify a = 8, b = 26, c = 15. Multiply a and c: 8 × 15 = 120. Find two numbers that multiply to 120 and add to 26: 20 and 6 (since 20 × 6 = 120 and 20 + 6 = 26).
    Full step-by-step solution

    Step 1: Identify a = 8, b = 26, c = 15. Step 2: Multiply a and c: 8 × 15 = 120. Step 3: Find two numbers that multiply to 120 and add to 26: 20 and 6 (since 20 × 6 = 120 and 20 + 6 = 26). Step 4: Rewrite the middle term: 8x² + 20x + 6x + 15. Step 5: Factor by grouping: (8x² + 20x) + (6x + 15). Step 6: Factor out common factors: 4x(2x + 5) + 3(2x + 5). Step 7: Factor out the common binomial: (2x + 5)(4x + 3). The answer is (4x + 3)(2x + 5).