Factor Polynomials

lessonbunny.com

Answer Key & Explanations

Factor Polynomials · Grade 9 · Worksheet 2

1. Charlotte is designing a rectangular mosaic where the area is represented by the polynomial 16x³ + 40x² + 24x square inches. The mosaic is composed of square tiles. If the height of the mosaic is (2x + 3) inches, what is the width of the mosaic in factored form? Answer: 8x² + 8x Solution: The area of a rectangle is width times height. So Area = Width × Height. We know Area = 16x³ + 40x² + 24x and Height = (2x + 3).
   Full step-by-step solution

   Step 1: The area of a rectangle is width times height. So Area = Width × Height. We know Area = 16x³ + 40x² + 24x and Height = (2x + 3). To find Width, we divide the area by the height: Width = (16x³ + 40x² + 24x) ÷ (2x + 3). Step 2: First, factor out the greatest common factor from the area polynomial. All terms have a factor of 8x: 16x³ + 40x² + 24x = 8x(2x² + 5x + 3). Step 3: Now factor the quadratic 2x² + 5x + 3. Find two numbers that multiply to 2 × 3 = 6 and add to 5. These numbers are 2 and 3. Rewrite the middle term: 2x² + 2x + 3x + 3. Group: (2x² + 2x) + (3x + 3) = 2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1). Step 4: So the area polynomial factors completely as: 16x³ + 40x² + 24x = 8x(2x + 3)(x + 1). Step 5: Since Area = Width × Height and Height = (2x + 3), we have: Width = [8x(2x + 3)(x + 1)] ÷ (2x + 3) = 8x(x + 1) = 8x² + 8x. The width of the mosaic in factored form is 8x² + 8x.

2. Factor completely: 12x³ - 27x² - 15x = ? Answer: 3x(4x + 1)(x - 5) Solution: Identify the greatest common factor of 12x³, -27x², and -15x. The GCF is 3x. Factor out 3x: 3x(4x² - 9x - 5) Factor the quadratic expression 4x² - 9x - 5 Find two numbers that multiply to (4)(-5) = -20 and add to -9.
   Full step-by-step solution

   Step 1: Identify the greatest common factor of 12x³, -27x², and -15x. The GCF is 3x. Step 2: Factor out 3x: 3x(4x² - 9x - 5) Step 3: Factor the quadratic expression 4x² - 9x - 5 Step 4: Find two numbers that multiply to (4)(-5) = -20 and add to -9. These numbers are -10 and 1. Step 5: Rewrite the middle term: 4x² - 10x + 1x - 5 Step 6: Factor by grouping: (4x² - 10x) + (1x - 5) = 2x(2x - 5) + 1(2x - 5) Step 7: Factor out the common binomial: (2x - 5)(2x + 1) Step 8: The complete factorization is: 3x(2x - 5)(2x + 1) The answer is 3x(2x - 5)(2x + 1).

3. A rectangular garden has an area represented by the polynomial expression 12x²y + 18xy² - 24xy. If the width of the garden is 6xy, what is the length of the garden in simplest form? Answer: 2x+3y-4 Solution: Area = 12x²y + 18xy² - 24xy Width = 6xy Length = ? Area = Length × Width So, Length = Area ÷ Width. Length = (12x²y + 18xy² - 24xy) ÷ (6xy) Divide each term of the polynomial by 6xy.
   Full step-by-step solution

   We are given: Area = 12x²y + 18xy² - 24xy Width = 6xy Length = ? Step 1: Recall the formula for the area of a rectangle: Area = Length × Width So, Length = Area ÷ Width. Step 2: Write the division: Length = (12x²y + 18xy² - 24xy) ÷ (6xy) Step 3: Divide each term of the polynomial by 6xy. First term: 12x²y ÷ (6xy) = (12/6) × (x²/x) × (y/y) = 2 × x¹ × 1 = 2x Second term: 18xy² ÷ (6xy) = (18/6) × (x/x) × (y²/y) = 3 × 1 × y¹ = 3y Third term: -24xy ÷ (6xy) = (-24/6) × (x/x) × (y/y) = -4 × 1 × 1 = -4 Step 4: Combine the results: Length = 2x + 3y - 4 Step 5: Check: Multiply Length × Width = (2x + 3y - 4) × (6xy) = 2x × 6xy = 12x²y + 3y × 6xy = 18xy² + (-4) × 6xy = -24xy This matches the given Area. Final answer: 2x + 3y - 4

4. Factor completely: 6x² - 13x - 5 = ? Answer: (2x - 5)(3x + 1) Solution: Identify coefficients. We have a = 6, b = -13, c = -5. Multiply a and c.
   Full step-by-step solution

   Let's factor 6x² - 13x - 5 completely. Step 1: Identify coefficients. We have a = 6, b = -13, c = -5. Step 2: Multiply a and c. a * c = 6 * (-5) = -30. Step 3: Find two numbers that multiply to -30 and add to b (-13). List factor pairs of -30: 1 and -30 → 1 + (-30) = -29 (no) -1 and 30 → -1 + 30 = 29 (no) 2 and -15 → 2 + (-15) = -13 (yes!) -2 and 15 → -2 + 15 = 13 (no) 3 and -10 → 3 + (-10) = -7 (no) 5 and -6 → 5 + (-6) = -1 (no) So the numbers are 2 and -15. Step 4: Rewrite the middle term using these numbers. 6x² - 13x - 5 = 6x² + 2x - 15x - 5. Step 5: Factor by grouping. Group the first two terms and the last two terms: (6x² + 2x) + (-15x - 5) Factor out the greatest common factor from each group: From (6x² + 2x), factor out 2x: 2x(3x + 1) From (-15x - 5), factor out -5: -5(3x + 1) Now we have: 2x(3x + 1) - 5(3x + 1) Step 6: Factor out the common binomial (3x + 1). (3x + 1)(2x - 5) Step 7: Write the final factored form. (2x - 5)(3x + 1) Therefore, the completely factored form is (2x - 5)(3x + 1).

5. 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

6. 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.

7. Factor completely: 54x⁵y³ - 81x⁴y⁴ = ? Answer: 27x⁴y³(2x - 3y) Solution: Find the GCF of the coefficients 54 and 81. The factors of 54 are 1,2,3,6,9,18,27,54. The factors of 81 are 1,3,9,27,81.
   Full step-by-step solution

   Step 1: Find the GCF of the coefficients 54 and 81. The factors of 54 are 1,2,3,6,9,18,27,54. The factors of 81 are 1,3,9,27,81. The greatest common factor is 27. Step 2: Find the GCF of the x terms: x⁵ and x⁴. The smallest exponent is 4, so x⁴ is common. Step 3: Find the GCF of the y terms: y³ and y⁴. The smallest exponent is 3, so y³ is common. Step 4: The overall GCF is 27x⁴y³. Step 5: Divide each term by 27x⁴y³: 54x⁵y³ ÷ 27x⁴y³ = 2x 81x⁴y⁴ ÷ 27x⁴y³ = 3y Step 6: Write the factored form: 27x⁴y³(2x - 3y) The answer is 27x⁴y³(2x - 3y).

8. A rectangular garden has an area represented by the polynomial 24x² + 60x square meters. The garden is divided into 3 equal rectangular sections of equal width along its length. A diagram shows a large rectangle labeled 'Garden Area = 24x² + 60x', with two vertical dashed lines dividing it into three equal-width sections, each labeled 'x' in width. The entire length of the garden is 3x meters. What is the width of the garden in meters, expressed as a polynomial in factored form? Answer: 4(2x + 5) Solution: Write the formula for area of a rectangle: Area = length x width. Substitute the given values: 24x² + 60x = (3x) x width. Solve for width by dividing both sides by 3x: width = (24x² + 60x) / (3x).
   Full step-by-step solution

   Step 1: Write the formula for area of a rectangle: Area = length x width. Step 2: Substitute the given values: 24x² + 60x = (3x) x width. Step 3: Solve for width by dividing both sides by 3x: width = (24x² + 60x) / (3x). Step 4: Divide each term in the numerator by 3x: 24x² / 3x = 8x, and 60x / 3x = 20. Step 5: So width = 8x + 20. Step 6: Factor out the greatest common factor from 8x + 20. The GCF of 8x and 20 is 4. Step 7: Factored form: 4(2x + 5). The width of the garden in factored form is 4(2x + 5).