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Factor Polynomials

Grade 9 · Algebra · Worksheet 2

  1. A rectangular prism has a volume represented by the polynomial 24x³y² - 36x²y³ + 48x⁴y. If the height of the prism is 12xy, what is the area of the base in simplest form? Answer: ______________
  2. Liam is designing a rectangular garden with an area represented by the polynomial expression 2x² + 7x + 6 square meters. He needs to determine the dimensions of the garden in terms of binomial factors. What are the length and width of Liam's garden when expressed as factors of this polynomial? Answer: ______________
  3. A manufacturing company is designing a new packaging box with a volume represented by the polynomial 4x³ + 18x² + 20x cubic centimeters. The height of the box is 2x cm, and the length is (x + 2) cm. What expression represents the width of the box in terms of x? Answer: ______________
  4. Factor completely: 27x³y² - 72x²y³ = ? Answer: ______________
  5. A rectangular garden has an area represented by the polynomial 6x² + 11x - 10 square meters. The length of the garden is 3x - 2 meters. What is the width of the garden in terms of x? Answer: ______________
  6. A manufacturing company is designing a new packaging box with a volume represented by the polynomial 12x³ + 8x² - 15x cubic centimeters. The engineers need to factor this expression completely to determine the possible dimensions of the box. What is the completely factored form of the volume polynomial? Answer: ______________
  7. Factor completely: 36x⁵y³ - 54x⁴y⁴ + 18x³y⁵ = ? Answer: ______________
  8. A rectangular garden has an area represented by the polynomial expression 2x² + 7x + 6 square meters. The length of the garden is (x + 2) meters. What is the width of the garden in terms of x? Answer: ______________
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Answer Key & Explanations

Factor Polynomials · Grade 9 · Worksheet 2

  1. A rectangular prism has a volume represented by the polynomial 24x³y² - 36x²y³ + 48x⁴y. If the height of the prism is 12xy, what is the area of the base in simplest form? Answer: 2x²y - 3xy² + 4x³ Solution: The volume of a rectangular prism is given by V = base area × height To find the base area, divide the volume by the height: (24x³y² - 36x²y³ + 48x⁴y) ÷ (12xy) - First term: 24x³y² ÷ 12xy = (24/12)x^(3-1)y^(2-1) = 2x²y - Second term: -36x²y³ ÷ 12xy = (-36/12)x^(2-1)y^(3-1) = -3xy² - Third term:…
    Full step-by-step solution

    Step 1: The volume of a rectangular prism is given by V = base area × height Step 2: To find the base area, divide the volume by the height: (24x³y² - 36x²y³ + 48x⁴y) ÷ (12xy) Step 3: Divide each term separately: - First term: 24x³y² ÷ 12xy = (24/12)x^(3-1)y^(2-1) = 2x²y - Second term: -36x²y³ ÷ 12xy = (-36/12)x^(2-1)y^(3-1) = -3xy² - Third term: 48x⁴y ÷ 12xy = (48/12)x^(4-1)y^(1-1) = 4x³ Step 4: Combine the results: 2x²y - 3xy² + 4x³ The area of the base is 2x²y - 3xy² + 4x³.

  2. Liam is designing a rectangular garden with an area represented by the polynomial expression 2x² + 7x + 6 square meters. He needs to determine the dimensions of the garden in terms of binomial factors. What are the length and width of Liam's garden when expressed as factors of this polynomial? Answer: (2x + 3)(x + 2) Solution: Identify the coefficients. We have a quadratic in the form ax² + bx + c, where: a = 2 b = 7 c = 6 Multiply a and c. a * c = 2 * 6 = 12 Find two numbers that multiply to 12 and add to b (which is 7).
    Full step-by-step solution

    Let's factor the polynomial 2x² + 7x + 6 step by step. Step 1: Identify the coefficients. We have a quadratic in the form ax² + bx + c, where: a = 2 b = 7 c = 6 Step 2: Multiply a and c. a * c = 2 * 6 = 12 Step 3: Find two numbers that multiply to 12 and add to b (which is 7). 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 4: Rewrite the middle term (7x) using these two numbers. 2x² + 7x + 6 = 2x² + 3x + 4x + 6 Step 5: Factor by grouping. Group the first two terms and the last two terms: (2x² + 3x) + (4x + 6) Factor out the greatest common factor from each group: From (2x² + 3x), factor out x: x(2x + 3) From (4x + 6), factor out 2: 2(2x + 3) Now we have: x(2x + 3) + 2(2x + 3) Step 6: Factor out the common binomial factor (2x + 3). (2x + 3)(x + 2) Step 7: Check by multiplying. (2x + 3)(x + 2) = 2x*x + 2x*2 + 3*x + 3*2 = 2x² + 4x + 3x + 6 = 2x² + 7x + 6 ✓ Conclusion: The length and width of the garden are (2x + 3) meters and (x + 2) meters (order does not matter).

  3. A manufacturing company is designing a new packaging box with a volume represented by the polynomial 4x³ + 18x² + 20x cubic centimeters. The height of the box is 2x cm, and the length is (x + 2) cm. What expression represents the width of the box in terms of x? Answer: 2x + 5 Solution: When working with three-dimensional objects, the volume is calculated by multiplying length, width, and height.
    Full step-by-step solution

    When working with three-dimensional objects, the volume is calculated by multiplying length, width, and height. If you know the volume and two of the dimensions, you can find the third dimension using polynomial division. This concept applies to many real-world situations like packaging design, container manufacturing, and architectural planning where dimensions are related through polynomial expressions.

  4. Factor completely: 27x³y² - 72x²y³ = ? Answer: 9x²y²(3x - 8y) Solution: Identify the GCF of the coefficients 27 and 72. The GCF is 9. Identify the GCF of the variable parts.
    Full step-by-step solution

    Step 1: Identify the GCF of the coefficients 27 and 72. The GCF is 9. Step 2: Identify the GCF of the variable parts. For x, the lowest power is x². For y, the lowest power is y². So the GCF of the variable parts is x²y². Step 3: The overall GCF is 9x²y². Step 4: Factor out the GCF from each term: 27x³y² ÷ 9x²y² = 3x -72x²y³ ÷ 9x²y² = -8y Step 5: Write the factored form: 9x²y²(3x - 8y) The completely factored form is 9x²y²(3x - 8y).

  5. A rectangular garden has an area represented by the polynomial 6x² + 11x - 10 square meters. 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 = length × width Area = 6x² + 11x - 10 Length = 3x - 2 We need to find the width. Write the equation for width.
    Full step-by-step solution

    We know the area of the rectangle is given by: Area = length × width Given: Area = 6x² + 11x - 10 Length = 3x - 2 We need to find the width. Step 1: Write the equation for width. Width = Area ÷ Length Width = (6x² + 11x - 10) ÷ (3x - 2) Step 2: Factor the area polynomial if possible, or perform polynomial long division. Let's try factoring: We look for two numbers that multiply to 6 × (-10) = -60 and add to 11. Those numbers are 15 and -4. Step 3: Rewrite the middle term 11x as 15x - 4x. 6x² + 15x - 4x - 10 Step 4: Factor by grouping. Group the first two terms: 3x(2x + 5) Group the last two terms: -2(2x + 5) Step 5: Factor out (2x + 5). (2x + 5)(3x - 2) So: 6x² + 11x - 10 = (2x + 5)(3x - 2) Step 6: Now compute width. Width = (2x + 5)(3x - 2) ÷ (3x - 2) Cancel (3x - 2) from numerator and denominator (valid as long as 3x - 2 ≠ 0). Step 7: We get: Width = 2x + 5 Final answer: The width is 2x + 5 meters.

  6. A manufacturing company is designing a new packaging box with a volume represented by the polynomial 12x³ + 8x² - 15x cubic centimeters. The engineers need to factor this expression completely to determine the possible dimensions of the box. What is the completely factored form of the volume polynomial? Answer: x(2x + 3)(6x - 5) Solution: Factoring polynomials involves breaking them down into simpler multiplicative components. This is similar to finding the prime factors of numbers, but with algebraic expressions.
    Full step-by-step solution

    Factoring polynomials involves breaking them down into simpler multiplicative components. This is similar to finding the prime factors of numbers, but with algebraic expressions. The process typically begins by identifying any common factors in all terms, then applying methods like grouping or recognizing special patterns to factor what remains. In real-world applications like packaging design, these factors often represent physical dimensions of objects.

  7. 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).

  8. A rectangular garden has an area represented by the polynomial expression 2x² + 7x + 6 square meters. The length of the garden is (x + 2) meters. What is the width of the garden in terms of x? Answer: (2x + 3) Solution: Area = length × width Area = 2x² + 7x + 6 Length = x + 2 Let width = W (x + 2) × W = 2x² + 7x + 6 To find W, divide the area polynomial by the length polynomial: Divide 2x² + 7x + 6 by x + 2 2x² ÷ x = 2x Multiply 2x by (x + 2): 2x² + 4x (2x² + 7x + 6) − (2x² + 4x) = (7x − 4x) + 6 = 3x + 6 3x ÷ x…
    Full step-by-step solution

    We know the area of the rectangle is given by: Area = length × width Given: Area = 2x² + 7x + 6 Length = x + 2 Let width = W So: (x + 2) × W = 2x² + 7x + 6 To find W, divide the area polynomial by the length polynomial: Step 1: Set up the division: Divide 2x² + 7x + 6 by x + 2 Step 2: Divide the first term: 2x² ÷ x = 2x Multiply 2x by (x + 2): 2x² + 4x Subtract from the original polynomial: (2x² + 7x + 6) − (2x² + 4x) = (7x − 4x) + 6 = 3x + 6 Step 3: Divide the next term: 3x ÷ x = 3 Multiply 3 by (x + 2): 3x + 6 Subtract: (3x + 6) − (3x + 6) = 0 Step 4: Conclusion: The quotient is 2x + 3, remainder 0. So the width is 2x + 3 meters. Final answer: 2x + 3