Factor Polynomials

lessonbunny.com

Answer Key & Explanations

Factor Polynomials · Grade 9 · Worksheet 1

1. Hana is designing a rectangular metal sheet for a sculpture. The area of the sheet is given by the polynomial 28x⁴ + 20x³ - 12x² square centimeters. She needs to factor out the greatest common factor (GCF) to simplify her design calculations. What is the factored form of this polynomial? Answer: 4x²(7x² + 5x - 3) Solution: Identify the coefficients: 28, 20, and -12. The GCF of these numbers is 4. Identify the variable parts: x⁴, x³, and x².
   Full step-by-step solution

   Step 1: Identify the coefficients: 28, 20, and -12. The GCF of these numbers is 4. Step 2: Identify the variable parts: x⁴, x³, and x². The smallest exponent is 2, so the GCF includes x². Step 3: The overall GCF is 4x². Step 4: Divide each term by 4x²: - 28x⁴ ÷ 4x² = 7x² - 20x³ ÷ 4x² = 5x - -12x² ÷ 4x² = -3 Step 5: Write the factored form: 4x²(7x² + 5x - 3).

2. Hana is designing a rectangular garden for a community project. The area of the garden is represented by the polynomial 16x² + 24x square meters, where x is a positive integer representing the width in meters. She needs to factor this polynomial to find the length and width expressions. What is the factored form of the polynomial 16x² + 24x? Answer: 8x(2x + 3) Solution: Identify the greatest common factor (GCF) of the coefficients 16 and 24. The GCF of 16 and 24 is 8. Identify the greatest common factor of the variable parts.
   Full step-by-step solution

   Step 1: Identify the greatest common factor (GCF) of the coefficients 16 and 24. The GCF of 16 and 24 is 8. Step 2: Identify the greatest common factor of the variable parts. Both terms have at least one factor of x, so the GCF for variables is x. Step 3: The overall GCF is 8x. Step 4: Divide each term by the GCF: 16x² ÷ 8x = 2x, and 24x ÷ 8x = 3. Step 5: Write the factored form as the GCF times the remaining terms: 8x(2x + 3). The factored form is 8x(2x + 3).

3. Liam is designing a rectangular garden for his community center. The area of the garden can be modeled by the polynomial expression 6x² + 11x - 10, where x represents the length in meters. To determine the optimal layout, Liam needs to factor this expression completely to find its binomial dimensions. What is the factored form of the area? Answer: (3x - 2)(2x + 5) Solution: We are factoring the quadratic expression: 6x² + 11x - 10. Multiply the coefficient of x² (which is 6) by the constant term (which is -10). 6 * (-10) = -60.
   Full step-by-step solution

   We are factoring the quadratic expression: 6x² + 11x - 10. Step 1: Multiply the coefficient of x² (which is 6) by the constant term (which is -10). 6 * (-10) = -60. Step 2: Find two numbers that multiply to -60 and add to the middle coefficient 11. List pairs of factors of -60: (1, -60) → 1 + (-60) = -59 (no) (-1, 60) → -1 + 60 = 59 (no) (2, -30) → 2 + (-30) = -28 (no) (-2, 30) → -2 + 30 = 28 (no) (3, -20) → 3 + (-20) = -17 (no) (-3, 20) → -3 + 20 = 17 (no) (4, -15) → 4 + (-15) = -11 (no) (-4, 15) → -4 + 15 = 11 (yes) So the numbers are -4 and 15. Step 3: Rewrite the middle term 11x as -4x + 15x. 6x² + 11x - 10 = 6x² - 4x + 15x - 10. Step 4: Factor by grouping. Group the first two terms and the last two terms: (6x² - 4x) + (15x - 10) Factor out the greatest common factor from each group: 2x(3x - 2) + 5(3x - 2) Step 5: Factor out the common binomial (3x - 2): (3x - 2)(2x + 5) Step 6: Check by expanding: (3x - 2)(2x + 5) = 3x*2x + 3x*5 - 2*2x - 2*5 = 6x² + 15x - 4x - 10 = 6x² + 11x - 10 ✓ Final factored form: (3x - 2)(2x + 5)

4. Factor completely: 16x³y² - 36x²y³ + 28x²y² = ? Answer: 4x²y²(4x - 9y + 7) Solution: Find the GCF of the coefficients: 16, 36, 28. The GCF is 4. Find the GCF of the x terms: x³, x², x².
   Full step-by-step solution

   Step 1: Find the GCF of the coefficients: 16, 36, 28. The GCF is 4. Step 2: Find the GCF of the x terms: x³, x², x². The smallest exponent is 2, so x² is common. Step 3: Find the GCF of the y terms: y², y³, y². The smallest exponent is 2, so y² is common. Step 4: The overall GCF is 4x²y². Step 5: Divide each term by 4x²y²: 16x³y² ÷ 4x²y² = 4x 36x²y³ ÷ 4x²y² = 9y 28x²y² ÷ 4x²y² = 7 Step 6: Write the factored form: 4x²y²(4x - 9y + 7) The answer is 4x²y²(4x - 9y + 7).

5. Mere is designing a rectangular community garden. The area of the garden is represented by the polynomial 18x² + 30x square meters. To order fencing materials, Mere needs to factor this expression by finding the greatest common factor (GCF). What is the factored form of the area of the garden? Answer: 6x(3x + 5) Solution: Identify the coefficients of each term: 18 and 30. Find the greatest common factor (GCF) of 18 and 30. The factors of 18 are 1, 2, 3, 6, 9, 18.
   Full step-by-step solution

   Step 1: Identify the coefficients of each term: 18 and 30. Step 2: Find the greatest common factor (GCF) of 18 and 30. The factors of 18 are 1, 2, 3, 6, 9, 18. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. The greatest common factor is 6. Step 3: Identify the variable part. The first term is 18x² and the second term is 30x. The smallest power of x common to both terms is x (since x² = x * x, and x = x * 1, so x is common). Step 4: The GCF of the entire polynomial is 6 * x = 6x. Step 5: Divide each term by the GCF: (18x²) / (6x) = 3x, and (30x) / (6x) = 5. Step 6: Write the factored form as the GCF times the remaining terms in parentheses: 6x(3x + 5). The answer is 6x(3x + 5).

6. A rectangular garden has an area represented by the polynomial expression 2x² + 7x + 6 square meters. The length of the garden is given as (2x + 3) meters. What is the width of the garden in terms of x? Answer: (x + 2) Solution: Area = length × width Area = 2x² + 7x + 6 Length = 2x + 3 We need to find the width in terms of x. Write the equation for width. Width = Area ÷ Length Width = (2x² + 7x + 6) ÷ (2x + 3) Factor the polynomial 2x² + 7x + 6.
   Full step-by-step solution

   We know the area of the rectangle is given by: Area = length × width Given: Area = 2x² + 7x + 6 Length = 2x + 3 We need to find the width in terms of x. Step 1: Write the equation for width. Width = Area ÷ Length Width = (2x² + 7x + 6) ÷ (2x + 3) Step 2: Factor the polynomial 2x² + 7x + 6. We look for two numbers that multiply to 2×6 = 12 and add to 7. Those numbers are 3 and 4. Step 3: Rewrite the middle term using 3 and 4. 2x² + 7x + 6 = 2x² + 3x + 4x + 6 Step 4: Factor by grouping. Group the first two terms: 2x² + 3x = x(2x + 3) Group the last two terms: 4x + 6 = 2(2x + 3) So: 2x² + 3x + 4x + 6 = x(2x + 3) + 2(2x + 3) Step 5: Factor out the common factor (2x + 3). (2x + 3)(x + 2) So: 2x² + 7x + 6 = (2x + 3)(x + 2) Step 6: Now substitute back into the width formula. Width = (2x² + 7x + 6) ÷ (2x + 3) = [(2x + 3)(x + 2)] ÷ (2x + 3) Step 7: Cancel the common factor (2x + 3) (assuming 2x + 3 ≠ 0). Width = x + 2 Final answer: The width is (x + 2) meters.

7. A construction company is designing a rectangular parking lot with an area represented by the polynomial 12x² - 7x - 10 square meters. The project manager needs to determine the binomial factors that represent the possible length and width dimensions for planning purposes. What are the length and width of the parking lot expressed as binomial factors of this polynomial? Answer: (4x - 5)(3x + 2) Solution: Factoring quadratic polynomials involves finding two binomials whose product equals the original polynomial.
   Full step-by-step solution

   Factoring quadratic polynomials involves finding two binomials whose product equals the original polynomial. This process requires identifying factors of the leading coefficient and constant term that combine to produce the middle coefficient. The factored form reveals the dimensions of rectangular areas in real-world applications like construction planning.