Factor Polynomials

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Factor Polynomials · Grade 9 · Worksheet 3

1. Factor completely: 27x⁴y² - 72x³y³ + 36x²y⁴ = ? Answer: 9x²y²(3x - 2y)(x - 2y) Solution: Find the GCF of the coefficients 27, 72, and 36. The GCF is 9. Find the GCF of the x terms: x⁴, x³, x².
   Full step-by-step solution

   Step 1: Find the GCF of the coefficients 27, 72, and 36. The GCF is 9. Step 2: Find the GCF of the x terms: x⁴, x³, x². The smallest exponent is 2, so the GCF is x². Step 3: Find the GCF of the y terms: y², y³, y⁴. The smallest exponent is 2, so the GCF is y². Step 4: The overall GCF is 9x²y². Step 5: Factor out the GCF from each term: 27x⁴y² ÷ 9x²y² = 3x² -72x³y³ ÷ 9x²y² = -8xy 36x²y⁴ ÷ 9x²y² = 4y² So we have: 9x²y²(3x² - 8xy + 4y²) Step 6: Now factor the trinomial 3x² - 8xy + 4y². This is a quadratic in x and y. Step 7: Find two numbers that multiply to (3)(4) = 12 and add to -8. The numbers are -6 and -2 (since -6 × -2 = 12 and -6 + -2 = -8). Step 8: Rewrite the middle term: 3x² - 6xy - 2xy + 4y² Step 9: Factor by grouping: 3x(x - 2y) - 2y(x - 2y) Step 10: Factor out (x - 2y): (3x - 2y)(x - 2y) Step 11: Combine all factors: 9x²y²(3x - 2y)(x - 2y) The completely factored form is 9x²y²(3x - 2y)(x - 2y).

2. Factor completely: 42x⁴y³ - 63x³y⁴ + 21x²y⁵ = ? Answer: 21x²y³(2x - y)(x - y) Solution: Find the GCF of the coefficients 42, 63, and 21. The GCF is 21. For the variable x, the smallest exponent is 2 (from x² in the third term).
   Full step-by-step solution

   Step 1: Find the GCF of the coefficients 42, 63, and 21. The GCF is 21. Step 2: For the variable x, the smallest exponent is 2 (from x² in the third term). So the GCF includes x². Step 3: For the variable y, the smallest exponent is 3 (from y³ in the first term). So the GCF includes y³. Step 4: The GCF is 21x²y³. Step 5: Factor out the GCF from each term: 42x⁴y³ ÷ 21x²y³ = 2x² -63x³y⁴ ÷ 21x²y³ = -3xy 21x²y⁵ ÷ 21x²y³ = y² So we have: 21x²y³(2x² - 3xy + y²) Step 6: Now factor the quadratic trinomial 2x² - 3xy + y². This is a quadratic in x (or y). Find two numbers that multiply to (2)(1) = 2 and add to -3. The numbers are -2 and -1. Step 7: Rewrite the middle term: 2x² - 2xy - xy + y² Step 8: Factor by grouping: 2x(x - y) - y(x - y) = (2x - y)(x - y) Step 9: Combine all factors: 21x²y³(2x - y)(x - y) The completely factored form is 21x²y³(2x - y)(x - y).

3. A rectangular community garden plot has an area represented by the polynomial 12x² - 7x - 10 square meters. The garden committee needs to install decorative fencing along the perimeter and must determine the dimensions in factored form. If the length is represented by (4x - 5) meters, what expression represents the width of the garden plot? Answer: (3x + 2) Solution: We know the area is 12x² - 7x - 10 and the length is (4x - 5). To find the width, we divide the area by the length: (12x² - 7x - 10) ÷ (4x - 5).
   Full step-by-step solution

   Step 1: We know the area is 12x² - 7x - 10 and the length is (4x - 5). Step 2: To find the width, we divide the area by the length: (12x² - 7x - 10) ÷ (4x - 5). Step 3: Factor the polynomial 12x² - 7x - 10 by looking for two numbers that multiply to (12)(-10) = -120 and add to -7. Step 4: The numbers -15 and 8 satisfy this: -15 × 8 = -120 and -15 + 8 = -7. Step 5: Rewrite the middle term: 12x² - 15x + 8x - 10. Step 6: Factor by grouping: (12x² - 15x) + (8x - 10) = 3x(4x - 5) + 2(4x - 5). Step 7: Factor out the common binomial: (4x - 5)(3x + 2). Step 8: Since the length is (4x - 5), the width must be (3x + 2). The width of the garden plot is (3x + 2) meters.

4. A rectangular garden has an area represented by the polynomial expression 6x² + 11x + 4 square meters. If the length of the garden is 2x + 1 meters, what is the width of the garden in terms of x? Answer: 3x + 4 Solution: Area = length × width Area = 6x² + 11x + 4 Length = 2x + 1 Width = ? (2x + 1) × width = 6x² + 11x + 4 width = (6x² + 11x + 4) ÷ (2x + 1) Factor the numerator if possible, or perform polynomial division.
   Full step-by-step solution

   We know the area of the rectangle is given by: Area = length × width Here: Area = 6x² + 11x + 4 Length = 2x + 1 Width = ? Step 1: Set up the equation (2x + 1) × width = 6x² + 11x + 4 So, width = (6x² + 11x + 4) ÷ (2x + 1) Step 2: Factor the numerator if possible, or perform polynomial division. We try factoring: 6x² + 11x + 4 Multiply the coefficient of x² (6) by the constant term (4): 6 × 4 = 24 Find two numbers that multiply to 24 and add to 11: 8 and 3 Step 3: Rewrite the middle term using 8 and 3: 6x² + 8x + 3x + 4 Step 4: Factor by grouping Group: (6x² + 8x) + (3x + 4) Factor each group: 2x(3x + 4) + 1(3x + 4) Step 5: Factor out the common binomial (3x + 4): (3x + 4)(2x + 1) So, 6x² + 11x + 4 = (3x + 4)(2x + 1) Step 6: Now substitute back into the width formula: width = [(3x + 4)(2x + 1)] ÷ (2x + 1) Step 7: Cancel the common factor (2x + 1) (since 2x + 1 ≠ 0 for general x): width = 3x + 4 Final answer: The width is 3x + 4 meters.

5. Factor completely: 24x⁴y² - 56x³y³ + 16x²y⁴ = ? Answer: 8x²y²(3x² - 7xy + 2y²) Solution: Find the GCF of the coefficients 24, -56, and 16. The GCF is 8. Find the GCF of the x powers: x⁴, x³, x².
   Full step-by-step solution

   Step 1: Find the GCF of the coefficients 24, -56, and 16. The GCF is 8. Step 2: Find the GCF of the x powers: x⁴, x³, x². The smallest exponent is 2, so x² is common. Step 3: Find the GCF of the y powers: y², y³, y⁴. The smallest exponent is 2, so y² is common. Step 4: The overall GCF is 8x²y². Step 5: Factor out 8x²y² from each term: 24x⁴y² ÷ 8x²y² = 3x² -56x³y³ ÷ 8x²y² = -7xy 16x²y⁴ ÷ 8x²y² = 2y² Step 6: Write the factored form: 8x²y²(3x² - 7xy + 2y²) The answer is 8x²y²(3x² - 7xy + 2y²).

6. Hana is painting a mural on a rectangular wall. The area of the wall is given by the polynomial expression 18x² + 24x square feet. The height of the wall is 6x feet. What is the width of the wall in factored form? Answer: 3x + 4 Solution: The area of a rectangle is given by Area = height × width. We know Area = 18x² + 24x and height = 6x. To find width, divide area by height: width = (18x² + 24x) ÷ (6x).
   Full step-by-step solution

   Step 1: The area of a rectangle is given by Area = height × width. Step 2: We know Area = 18x² + 24x and height = 6x. Step 3: To find width, divide area by height: width = (18x² + 24x) ÷ (6x). Step 4: Factor the area polynomial: 18x² + 24x = 6x(3x + 4). Step 5: Now divide: width = [6x(3x + 4)] ÷ (6x). Step 6: Cancel the common factor 6x: width = 3x + 4. The width of the wall in factored form is 3x + 4.

7. Factor completely: 6x³ - 24x² - 30x Answer: 6x(x - 5)(x + 1) Solution: 6x³ - 24x² - 30x All terms have a common factor of 6x: 6x³ ÷ 6x = x² -24x² ÷ 6x = -4x -30x ÷ 6x = -5 6x³ - 24x² - 30x = 6x(x² - 4x - 5) We now factor x² - 4x - 5.
   Full step-by-step solution

   Let's factor the expression step by step. We start with: 6x³ - 24x² - 30x --- **Step 1: Look for a greatest common factor (GCF)** All terms have a common factor of 6x: 6x³ ÷ 6x = x² -24x² ÷ 6x = -4x -30x ÷ 6x = -5 So: 6x³ - 24x² - 30x = 6x(x² - 4x - 5) --- **Step 2: Factor the quadratic trinomial inside** We now factor x² - 4x - 5. We look for two numbers that multiply to -5 and add to -4. Possible pairs for -5: 1 and -5 → 1 + (-5) = -4 ✓ -1 and 5 → -1 + 5 = 4 ✗ So the correct pair is 1 and -5. Thus: x² - 4x - 5 = (x - 5)(x + 1) --- **Step 3: Write the complete factored form** Putting it all together: 6x³ - 24x² - 30x = 6x(x - 5)(x + 1) --- **Final Answer:** 6x(x - 5)(x + 1)

8. A construction company is designing a rectangular parking lot with an area represented by the polynomial 12x² + 29x + 15 square meters. The project manager needs to determine the dimensions in factored form to calculate the required fencing material. If the length is given as (4x + 3) meters, what is the width of the parking lot in terms of x? Answer: (3x + 5) Solution: The area is 12x² + 29x + 15 and the length is (4x + 3) Width = Area ÷ Length = (12x² + 29x + 15) ÷ (4x + 3) Use polynomial division or factoring to find the other factor Look for factors: (4x + 3)(?x + ?) = 12x² + 29x + 15 4x × ?x = 12x², so ?
   Full step-by-step solution

   Step 1: The area is 12x² + 29x + 15 and the length is (4x + 3) Step 2: Width = Area ÷ Length = (12x² + 29x + 15) ÷ (4x + 3) Step 3: Use polynomial division or factoring to find the other factor Step 4: Look for factors: (4x + 3)(?x + ?) = 12x² + 29x + 15 Step 5: 4x × ?x = 12x², so ? = 3 Step 6: 3 × ? = 15, so ? = 5 Step 7: Check: (4x + 3)(3x + 5) = 12x² + 20x + 9x + 15 = 12x² + 29x + 15 Step 8: The width is (3x + 5) meters