Factor Polynomials

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Factor Polynomials · Grade 9 · Worksheet 3

1. A manufacturing company is designing a new storage container with a volume represented by the polynomial 4x³ - 8x² - 12x cubic feet. The engineers need to factor out the greatest common factor to determine the base dimensions. What is the greatest common factor of the polynomial 4x³ - 8x² - 12x? Answer: 4x Solution: Step 1: Identify the terms of the polynomial: 4x³, -8x², and -12x Step 2: Find the GCF of the coefficients: 4, 8, and 12 - Factors of 4: 1, 2, 4 - Factors of 8: 1, 2, 4, 8 - Factors of 12: 1, 2, 3, 4, 6, 12 - The GCF of 4, 8, and 12 is 4 Step 3: Find the GCF of the variable parts: x³, x², and x…
   Full step-by-step solution

   Step 1: Identify the terms of the polynomial: 4x³, -8x², and -12x Step 2: Find the GCF of the coefficients: 4, 8, and 12 - Factors of 4: 1, 2, 4 - Factors of 8: 1, 2, 4, 8 - Factors of 12: 1, 2, 3, 4, 6, 12 - The GCF of 4, 8, and 12 is 4 Step 3: Find the GCF of the variable parts: x³, x², and x - The smallest power of x is x¹ - The GCF of the variable parts is x Step 4: Combine the numerical and variable GCFs: 4 × x = 4x Step 5: Verify by dividing each term by 4x: - 4x³ ÷ 4x = x² - -8x² ÷ 4x = -2x - -12x ÷ 4x = -3 - The result is x² - 2x - 3, confirming 4x is the GCF The greatest common factor is 4x.

2. A manufacturing company is designing a new storage container with a volume represented by the polynomial 12x³ - 3x² - 27x cubic centimeters. The engineers need to factor out the greatest common factor to determine the base dimensions for standard packaging. What is the greatest common factor of this polynomial? Answer: 3x Solution: The greatest common factor (GCF) of a polynomial is the largest monomial that divides each term of the polynomial.
   Full step-by-step solution

   The greatest common factor (GCF) of a polynomial is the largest monomial that divides each term of the polynomial. To find it, identify the GCF of the numerical coefficients and the smallest power of each variable that appears in all terms. Factoring out the GCF simplifies polynomial expressions and is often the first step in more complex factoring problems.

3. Mere is designing a rectangular metal frame. The area of the frame is given by the polynomial 28x³ + 21x² − 14x square centimeters. If the width of the frame is 7x centimeters, write a fully factored expression for the length of the frame. Answer: 4x² + 3x − 2 Solution: The area of a rectangle is length times width. So length = area / width. Area = 28x³ + 21x² − 14x, width = 7x.
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width. So length = area / width. Step 2: Area = 28x³ + 21x² − 14x, width = 7x. Step 3: Divide each term of the area by 7x: 28x³ / 7x = 4x², 21x² / 7x = 3x, (−14x) / 7x = −2. Step 4: The length is 4x² + 3x − 2. Step 5: Check if this can be factored: The GCF of 4x², 3x, and −2 is 1 (no common factor other than 1), so it is already fully factored. The fully factored expression for the length is 4x² + 3x − 2.

4. Factor completely: 36x⁵y³ - 54x⁴y⁴ + 18x³y⁵ = ? Answer: 18x³y³(2x² - 3xy + y²) Solution: Find the GCF of the coefficients 36, 54, and 18. The GCF is 18. For the variable x, the lowest exponent among x⁵, x⁴, and x³ is x³.
   Full step-by-step solution

   Step 1: Find the GCF of the coefficients 36, 54, and 18. The GCF is 18. Step 2: For the variable x, the lowest exponent among x⁵, x⁴, and x³ is x³. Step 3: For the variable y, the lowest exponent among y³, y⁴, and y⁵ is y³. Step 4: The GCF of the entire expression is 18x³y³. Step 5: Factor out the GCF from each term: 36x⁵y³ ÷ 18x³y³ = 2x² -54x⁴y⁴ ÷ 18x³y³ = -3xy 18x³y⁵ ÷ 18x³y³ = y² Step 6: Write the factored form: 18x³y³(2x² - 3xy + y²) The answer is 18x³y³(2x² - 3xy + y²).

5. Factor completely: 16x⁴ - 81 = ? Answer: (4x² + 9)(2x + 3)(2x - 3) Solution: Recognize that 16x⁴ - 81 is a difference of squares 16x⁴ = (4x²)² and 81 = 9² So 16x⁴ - 81 = (4x²)² - 9² Apply the difference of squares formula: a² - b² = (a + b)(a - b) (4x² + 9)(4x² - 9) Notice that 4x² - 9 is also a difference of squares 4x² = (2x)² and 9 = 3² So 4x² - 9 = (2x)² - 3² = (2x +…
   Full step-by-step solution

   Step 1: Recognize that 16x⁴ - 81 is a difference of squares 16x⁴ = (4x²)² and 81 = 9² So 16x⁴ - 81 = (4x²)² - 9² Step 2: Apply the difference of squares formula: a² - b² = (a + b)(a - b) (4x² + 9)(4x² - 9) Step 3: Notice that 4x² - 9 is also a difference of squares 4x² = (2x)² and 9 = 3² So 4x² - 9 = (2x)² - 3² = (2x + 3)(2x - 3) Step 4: The factor 4x² + 9 cannot be factored further over the real numbers Step 5: Write the complete factorization (4x² + 9)(2x + 3)(2x - 3) The answer is (4x² + 9)(2x + 3)(2x - 3).

6. 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 need to 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 inside the parentheses** We need to factor x² - 4x - 5. 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 factorization** Substitute back: 6x (x² - 4x - 5) = 6x (x - 5)(x + 1) --- **Final Answer:** 6x(x - 5)(x + 1)

7. 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.

8. Factor completely: 56x⁵y³ - 84x⁴y⁴ + 28x³y⁵ = ? Answer: 28x³y³(2x - y)(x - y) Solution: Find the GCF of the coefficients 56, 84, and 28. The GCF is 28. Find the GCF of the x terms: x⁵, x⁴, x³.
   Full step-by-step solution

   Step 1: Find the GCF of the coefficients 56, 84, and 28. The GCF is 28. Step 2: Find the GCF of the x terms: x⁵, x⁴, x³. The smallest exponent is 3, so x³ is common. Step 3: Find the GCF of the y terms: y³, y⁴, y⁵. The smallest exponent is 3, so y³ is common. Step 4: The overall GCF is 28x³y³. Step 5: Divide each term by 28x³y³: 56x⁵y³ ÷ 28x³y³ = 2x² 84x⁴y⁴ ÷ 28x³y³ = 3xy 28x³y⁵ ÷ 28x³y³ = y² So we have: 28x³y³(2x² - 3xy + y²) Step 6: Factor the trinomial 2x² - 3xy + y². This is a quadratic in x and y. Step 7: Find two numbers that multiply to (2)(1) = 2 and add to -3. The numbers are -2 and -1 (since -2 × -1 = 2 and -2 + -1 = -3). Step 8: Rewrite the middle term: 2x² - 2xy - xy + y² Step 9: Factor by grouping: 2x(x - y) - y(x - y) Step 10: Factor out (x - y): (2x - y)(x - y) Step 11: Combine all factors: 28x³y³(2x - y)(x - y) The completely factored form is 28x³y³(2x - y)(x - y).

9. 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).