Factor Trinomials

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Answer Key & Explanations

Factor Trinomials · Grade 9 · Worksheet 1

1. Factor: 27x² + 57x + 28 Answer: (3x + 4)(9x + 7) Solution: Identify a = 27, b = 57, c = 28. Multiply a and c: 27 × 28 = 756. Find two numbers that multiply to 756 and add to 57.
   Full step-by-step solution

   Step 1: Identify a = 27, b = 57, c = 28. Step 2: Multiply a and c: 27 × 28 = 756. Step 3: Find two numbers that multiply to 756 and add to 57. The factor pairs of 756 include 36 and 21 (since 36 × 21 = 756 and 36 + 21 = 57). Step 4: Rewrite the middle term: 27x² + 36x + 21x + 28. Step 5: Factor by grouping: (27x² + 36x) + (21x + 28). Step 6: Factor out common factors: 9x(3x + 4) + 7(3x + 4). Step 7: Factor out the common binomial: (3x + 4)(9x + 7). The answer is (3x + 4)(9x + 7).

2. A rectangular garden has an area represented by the expression 2x² + 7x + 6 square meters. The length and width are binomial expressions with integer coefficients. If the length is longer than the width, what is the factored form that represents the dimensions of the garden? Answer: (2x+3)(x+2) Solution: We are given the area: 2x² + 7x + 6. We need to factor this into two binomials (length and width) with integer coefficients, where length > width. The quadratic is: 2x² + 7x + 6 Here: a = 2, b = 7, c = 6.
   Full step-by-step solution

   We are given the area: 2x² + 7x + 6. We need to factor this into two binomials (length and width) with integer coefficients, where length > width. --- **Step 1: Identify a, b, c for factoring by grouping** The quadratic is: 2x² + 7x + 6 Here: a = 2, b = 7, c = 6. --- **Step 2: Multiply a and c** a * c = 2 * 6 = 12. We need two numbers that multiply to 12 and add to b = 7. --- **Step 3: Find the pair of numbers** List factor pairs of 12: (1, 12) → sum 13 (2, 6) → sum 8 (3, 4) → sum 7 ← this works. So the numbers are 3 and 4. --- **Step 4: Rewrite the middle term using these numbers** 2x² + 7x + 6 = 2x² + 3x + 4x + 6 --- **Step 5: Factor by grouping** Group terms: (2x² + 3x) + (4x + 6) Factor each group: x(2x + 3) + 2(2x + 3) --- **Step 6: Factor out the common binomial** (2x + 3)(x + 2) --- **Step 7: Identify length and width** The factors are (2x + 3) and (x + 2). Since 2x + 3 > x + 2 for positive x, length = 2x + 3, width = x + 2. --- **Final answer:** (2x + 3)(x + 2)

3. A rectangular garden has an area represented by the polynomial expression 2x² + 11x + 12 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 + 4) Solution: We are given the quadratic expression: 2x² + 11x + 12. We want to factor it into two binomials with integer coefficients: (ax + b)(cx + d). a = 2, b = 11, c = 12.
   Full step-by-step solution

   We are given the quadratic expression: 2x² + 11x + 12. We want to factor it into two binomials with integer coefficients: (ax + b)(cx + d). --- **Step 1: Multiply a and c** In the general form ax² + bx + c, here: a = 2, b = 11, c = 12. We need ac = 2 × 12 = 24. --- **Step 2: Find two numbers that multiply to ac and add to b** We need two integers whose product is 24 and whose sum is 11. Possible pairs for 24: (1, 24) → sum 25 (2, 12) → sum 14 (3, 8) → sum 11 ← this works. So the numbers are 3 and 8. --- **Step 3: Rewrite the middle term using these numbers** 2x² + 11x + 12 = 2x² + 3x + 8x + 12. --- **Step 4: Factor by grouping** Group the terms: (2x² + 3x) + (8x + 12) Factor each group: x(2x + 3) + 4(2x + 3) --- **Step 5: Factor out the common binomial** (2x + 3)(x + 4) --- **Step 6: Check by expanding** (2x + 3)(x + 4) = 2x(x) + 2x(4) + 3(x) + 3(4) = 2x² + 8x + 3x + 12 = 2x² + 11x + 12 ✓ --- **Final answer:** (2x + 3)(x + 4)

4. Factor: 4x² + 16x + 15 = ? Answer: (2x + 3)(2x + 5) Solution: Multiply the coefficient of x² (4) and the constant term (15): 4 × 15 = 60 Find two numbers that multiply to 60 and add to 16: 6 and 10 Rewrite the middle term using these numbers: 4x² + 6x + 10x + 15 Factor by grouping: (4x² + 6x) + (10x + 15) = 2x(2x + 3) + 5(2x + 3) Factor out the common…
   Full step-by-step solution

   Step 1: Multiply the coefficient of x² (4) and the constant term (15): 4 × 15 = 60 Step 2: Find two numbers that multiply to 60 and add to 16: 6 and 10 Step 3: Rewrite the middle term using these numbers: 4x² + 6x + 10x + 15 Step 4: Factor by grouping: (4x² + 6x) + (10x + 15) = 2x(2x + 3) + 5(2x + 3) Step 5: Factor out the common binomial: (2x + 3)(2x + 5) The answer is (2x + 3)(2x + 5).

5. A rectangular solar panel has an area represented by the quadratic expression 2x² + 11x + 12 square meters. The panel is divided into four smaller rectangular sections arranged in a 2×2 grid pattern, where the length and width of each section correspond to the binomial factors of this expression. Using the AC method for factoring quadratic trinomials, determine the factored form that represents the dimensions of this solar panel. Answer: (2x+3)(x+4) Solution: Identify a, b, and c from the quadratic expression 2x² + 11x + 12, where a=2, b=11, c=12. Multiply a and c: 2 × 12 = 24. Find two numbers that multiply to 24 and add to 11: 3 and 8 (since 3×8=24 and 3+8=11).
   Full step-by-step solution

   Step 1: Identify a, b, and c from the quadratic expression 2x² + 11x + 12, where a=2, b=11, c=12. Step 2: Multiply a and c: 2 × 12 = 24. Step 3: Find two numbers that multiply to 24 and add to 11: 3 and 8 (since 3×8=24 and 3+8=11). Step 4: Rewrite the middle term using these numbers: 2x² + 3x + 8x + 12. Step 5: Factor by grouping: (2x² + 3x) + (8x + 12) = x(2x+3) + 4(2x+3). Step 6: Factor out the common binomial: (2x+3)(x+4). The factored form is (2x+3)(x+4).

6. A rectangular garden has an area of (6x² + 19x + 10) square meters. If the length of the garden is (3x + 2) meters, what is the width of the garden in terms of x? Answer: (2x + 5) Solution: Area = 6x² + 19x + 10 Length = 3x + 2 Width = ? Recall the formula for the area of a rectangle. Area = Length × Width So, Width = Area ÷ Length Substitute the given expressions.
   Full step-by-step solution

   We are given: Area = 6x² + 19x + 10 Length = 3x + 2 Width = ? Step 1: Recall the formula for the area of a rectangle. Area = Length × Width So, Width = Area ÷ Length Step 2: Substitute the given expressions. Width = (6x² + 19x + 10) ÷ (3x + 2) Step 3: Factor the numerator if possible, since we suspect it is divisible by (3x + 2). We need two numbers that multiply to 6x² × 10 = 60x² and add to 19x. The numbers 15x and 4x work because 15x × 4x = 60x² and 15x + 4x = 19x. Step 4: Rewrite the middle term using 15x and 4x. 6x² + 19x + 10 = 6x² + 15x + 4x + 10 Step 5: Factor by grouping. Group the first two terms: 6x² + 15x = 3x(2x + 5) Group the last two terms: 4x + 10 = 2(2x + 5) Step 6: Factor out the common factor (2x + 5). 3x(2x + 5) + 2(2x + 5) = (2x + 5)(3x + 2) Step 7: Now substitute back into the Width expression. Width = [(2x + 5)(3x + 2)] ÷ (3x + 2) Step 8: Cancel the common factor (3x + 2) (assuming x is such that 3x + 2 ≠ 0). Width = 2x + 5 Final answer: 2x + 5 meters

7. A physics class is designing a projectile motion experiment where the height of a ball thrown upward is modeled by the quadratic equation h(t) = -3t² + 17t + 10, where h is height in meters and t is time in seconds. To analyze when the ball hits the ground, factor this quadratic expression using the AC method. Answer: (3t + 1)(-t + 10) Solution: In projectile motion problems, factoring quadratic equations helps determine key events like when an object hits the ground.
   Full step-by-step solution

   In projectile motion problems, factoring quadratic equations helps determine key events like when an object hits the ground. The AC method involves multiplying the leading coefficient and constant term, then finding factor pairs that sum to the middle coefficient. This technique works for any quadratic expression and reveals important information about the mathematical model.

8. A rectangular garden has an area represented by the quadratic expression x² + 7x + 12 square meters. If the length and width of this garden are both binomial expressions with integer coefficients, what are the dimensions of the garden? Answer: (x+3)(x+4) Solution: We are given the area of the rectangular garden as: x² + 7x + 12 square meters. The length and width are binomials with integer coefficients, so we need to factor the quadratic expression.
   Full step-by-step solution

   We are given the area of the rectangular garden as: x² + 7x + 12 square meters. The length and width are binomials with integer coefficients, so we need to factor the quadratic expression. --- **Step 1: Write the general factored form** We want: x² + 7x + 12 = (x + a)(x + b) where a and b are integers. --- **Step 2: Expand (x + a)(x + b)** (x + a)(x + b) = x² + (a + b)x + a*b. So comparing with x² + 7x + 12: a + b = 7 a * b = 12 --- **Step 3: Find integer pairs (a, b) whose product is 12** Possible integer pairs: (1, 12) → sum = 13 (no) (2, 6) → sum = 8 (no) (3, 4) → sum = 7 (yes) Also (4, 3) is the same pair essentially. --- **Step 4: Write the factorization** So a = 3, b = 4. Thus: x² + 7x + 12 = (x + 3)(x + 4). --- **Step 5: Interpret the dimensions** The length and width of the garden are (x + 3) meters and (x + 4) meters (order doesn't matter for dimensions). --- **Final Answer:** The dimensions are (x + 3) and (x + 4).