Factor Polynomials

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Factor Polynomials · Grade 9 · Worksheet 1

1. Noah is designing a geometric pattern on a coordinate grid. The pattern consists of a series of rectangles. The first rectangle has a width represented by the monomial 6x and a length represented by the binomial (x + 1). The area of this rectangle is 6x² + 6x square units. The second rectangle has a width of 6x and a length of (2x + 3), giving it an area of 12x² + 18x square units. If Noah combines these two rectangles into a single larger rectangle by placing them side by side along their widths, what is the total combined area of the larger rectangle, factored completely by its greatest common factor? Answer: 6x(3x + 4) Solution: Find the total area by adding the areas of the two rectangles: (6x² + 6x) + (12x² + 18x) = 18x² + 24x. Identify the greatest common factor (GCF) of the coefficients 18 and 24. The GCF of 18 and 24 is 6.
   Full step-by-step solution

   Step 1: Find the total area by adding the areas of the two rectangles: (6x² + 6x) + (12x² + 18x) = 18x² + 24x. Step 2: Identify the greatest common factor (GCF) of the coefficients 18 and 24. The GCF of 18 and 24 is 6. Step 3: Identify the GCF of the variable parts. Both terms have at least one factor of x, so the GCF includes x. Step 4: The overall GCF is 6x. Step 5: Factor 6x out of each term: 18x² ÷ 6x = 3x, and 24x ÷ 6x = 4. Step 6: Write the factored expression: 6x(3x + 4). The total combined area of the larger rectangle, factored completely, is 6x(3x + 4).

2. Liam is designing a rectangular garden with an area represented by the polynomial expression 3x² + 15x + 18 square feet. He needs to determine the greatest common factor of this expression to help him plan equal planting sections. What is the greatest common factor of the polynomial 3x² + 15x + 18? Answer: 3(x² + 5x + 6) Solution: Identify the coefficients of each term. The terms are: 3x², 15x, and 18. Their coefficients are: 3, 15, and 18.
   Full step-by-step solution

   Let's find the greatest common factor (GCF) of the polynomial 3x² + 15x + 18 step by step. Step 1: Identify the coefficients of each term. The terms are: 3x², 15x, and 18. Their coefficients are: 3, 15, and 18. Step 2: Find the GCF of the coefficients 3, 15, and 18. List the factors: - Factors of 3: 1, 3 - Factors of 15: 1, 3, 5, 15 - Factors of 18: 1, 2, 3, 6, 9, 18 The largest number common to all lists is 3. So, the GCF of the coefficients is 3. Step 3: Check if there is a common variable factor in all terms. The terms are 3x², 15x, and 18. The last term (18) has no x. Since not all terms have the variable x, the GCF does not include x. Step 4: Factor out the GCF 3 from the polynomial. Write each term divided by 3: 3x² ÷ 3 = x² 15x ÷ 3 = 5x 18 ÷ 3 = 6 So, 3x² + 15x + 18 = 3(x² + 5x + 6) Step 5: Verify by distributing. 3(x² + 5x + 6) = 3x² + 15x + 18, which matches the original polynomial. Therefore, the greatest common factor of 3x² + 15x + 18 is 3, and the factored form is 3(x² + 5x + 6).

3. Factor completely: 15x³ - 35x² - 30x = ? Answer: 5x(3x + 2)(x - 3) Solution: Identify the greatest common factor of 15x³, -35x², and -30x The GCF is 5x, so factor it out: 5x(3x² - 7x - 6) Now factor the quadratic expression 3x² - 7x - 6 Find two numbers that multiply to (3)(-6) = -18 and add to -7 The numbers are -9 and 2 (since -9 × 2 = -18 and -9 + 2 = -7) Rewrite the…
   Full step-by-step solution

   Step 1: Identify the greatest common factor of 15x³, -35x², and -30x Step 2: The GCF is 5x, so factor it out: 5x(3x² - 7x - 6) Step 3: Now factor the quadratic expression 3x² - 7x - 6 Step 4: Find two numbers that multiply to (3)(-6) = -18 and add to -7 Step 5: The numbers are -9 and 2 (since -9 × 2 = -18 and -9 + 2 = -7) Step 6: Rewrite the middle term: 3x² - 9x + 2x - 6 Step 7: Factor by grouping: 3x(x - 3) + 2(x - 3) Step 8: Factor out (x - 3): (3x + 2)(x - 3) Step 9: Combine all factors: 5x(3x + 2)(x - 3) The completely factored form is 5x(3x + 2)(x - 3).

4. Factor completely: 24x⁴y² - 40x³y³ + 16x²y⁴ = ? Answer: 8x²y²(3x² - 5xy + 2y²) Solution: Find the GCF of the coefficients 24, -40, and 16. The GCF is 8. Find the GCF of the x terms: x⁴, x³, x².
   Full step-by-step solution

   Step 1: Find the GCF of the coefficients 24, -40, and 16. The GCF is 8. Step 2: Find the GCF of the x terms: x⁴, x³, x². The smallest exponent is 2, so x² is common. Step 3: Find the GCF of the y terms: 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 -40x³y³ ÷ 8x²y² = -5xy 16x²y⁴ ÷ 8x²y² = 2y² Step 6: Write the factored form: 8x²y²(3x² - 5xy + 2y²) Step 7: Check if the trinomial 3x² - 5xy + 2y² can be factored further. It factors as (3x - 2y)(x - y). Step 8: The complete factorization is 8x²y²(3x - 2y)(x - y). The answer is 8x²y²(3x - 2y)(x - y).

5. Liam is designing a rectangular garden with an area represented by the polynomial expression 2x² + 11x + 12 square meters. He needs to determine the binomial factors that represent the possible length and width of the garden. What are the dimensions of Liam's garden in factored form? Answer: (2x+3)(x+4) Solution: We are given the area of the rectangular garden as: 2x² + 11x + 12 square meters. We need to factor this trinomial into two binomials (length and width).
   Full step-by-step solution

   We are given the area of the rectangular garden as: 2x² + 11x + 12 square meters. We need to factor this trinomial into two binomials (length and width). --- **Step 1: Identify coefficients** The trinomial is: 2x² + 11x + 12 Coefficients: a = 2, b = 11, c = 12 --- **Step 2: Factor by grouping (AC method)** Multiply a * c = 2 * 12 = 24. We need two numbers that multiply to 24 and add to b = 11. List factor pairs of 24: 1 and 24 → sum 25 (no) 2 and 12 → sum 14 (no) 3 and 8 → sum 11 (yes) So the numbers are 3 and 8. --- **Step 3: Rewrite the middle term** Split 11x into 3x + 8x: 2x² + 3x + 8x + 12 --- **Step 4: Factor by grouping** Group terms: (2x² + 3x) + (8x + 12) Factor out the GCF from each group: x(2x + 3) + 4(2x + 3) --- **Step 5: Factor out the common binomial** (2x + 3) is common: (2x + 3)(x + 4) --- **Step 6: Check** Multiply: (2x + 3)(x + 4) = 2x*x + 2x*4 + 3*x + 3*4 = 2x² + 8x + 3x + 12 = 2x² + 11x + 12 ✓ --- **Final answer:** The dimensions in factored form are (2x + 3) meters and (x + 4) meters.

6. Mere is designing a rectangular prism sculpture where the volume is represented by the polynomial expression 24x³ + 36x² + 12x cubic units. The height of the prism is 4x units. If the base is a rectangle, factor the expression representing the area of the base completely by taking out the greatest common factor. Answer: 6x² + 9x + 3 Solution: Volume of a rectangular prism = Base Area * Height. So Base Area = Volume / Height. Volume = 24x³ + 36x² + 12x, Height = 4x.
   Full step-by-step solution

   Step 1: Volume of a rectangular prism = Base Area * Height. So Base Area = Volume / Height. Step 2: Volume = 24x³ + 36x² + 12x, Height = 4x. Step 3: Divide each term of the volume by 4x: 24x³ / 4x = 6x² 36x² / 4x = 9x 12x / 4x = 3 Step 4: So Base Area = 6x² + 9x + 3. Step 5: Now factor out the greatest common factor from 6x² + 9x + 3. The GCF of 6, 9, and 3 is 3. There is no common variable factor since the last term is a constant (3). Step 6: Factor out 3: 6x² + 9x + 3 = 3(2x² + 3x + 1). The base area in factored form is 3(2x² + 3x + 1).

7. A rectangular garden has an area represented by the polynomial expression 2x² + 7x + 6 square meters. If the length of the garden is (2x + 3) meters, what expression represents the width of the garden in meters? Answer: (x + 2) Solution: Area = length × width Area = 2x² + 7x + 6 Length = 2x + 3 We need to find the width. Width = Area / Length Width = (2x² + 7x + 6) / (2x + 3) Factor the numerator 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. Step 1: Write the equation for width: Width = Area / Length Width = (2x² + 7x + 6) / (2x + 3) Step 2: Factor the numerator 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) Thus: 2x² + 7x + 6 = (2x + 3)(x + 2) Step 6: Now substitute back into the width expression: Width = (2x² + 7x + 6) / (2x + 3) = [(2x + 3)(x + 2)] / (2x + 3) Step 7: Cancel the common factor (2x + 3) (since 2x + 3 ≠ 0 for general x): Width = x + 2 Final answer: The width is (x + 2) meters.