Polynomial Multiplication

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Polynomial Multiplication · Grade 9 · Worksheet 2

1. A square patio has a side length of (2x + 7) feet. A circular fountain with a radius of (x - 2) feet is installed at the center of the patio. What is the area of the patio that is NOT covered by the fountain? Express your answer as a polynomial in standard form. (Use π = 3.14 and leave π as a symbol in the final expression.) Answer: 4x^2 + 28x + 49 - π(x^2 - 4x + 4) Solution: Find the area of the square patio. Area = side × side = (2x + 7)(2x + 7). Multiply using FOIL: (2x)(2x) = 4x^2, (2x)(7) = 14x, (7)(2x) = 14x, (7)(7) = 49.
   Full step-by-step solution

   Step 1: Find the area of the square patio. Area = side × side = (2x + 7)(2x + 7). Step 2: Multiply using FOIL: (2x)(2x) = 4x^2, (2x)(7) = 14x, (7)(2x) = 14x, (7)(7) = 49. Combine like terms: 4x^2 + (14x + 14x) + 49 = 4x^2 + 28x + 49. Step 3: Find the area of the circular fountain. Area = π × r^2 = π(x - 2)^2. Step 4: Multiply (x - 2)(x - 2): (x)(x) = x^2, (x)(-2) = -2x, (-2)(x) = -2x, (-2)(-2) = 4. Combine: x^2 - 4x + 4. So fountain area = π(x^2 - 4x + 4). Step 5: Subtract the fountain area from the patio area: (4x^2 + 28x + 49) - π(x^2 - 4x + 4). Step 6: Distribute the subtraction: 4x^2 + 28x + 49 - πx^2 + 4πx - 4π. Step 7: Since π is an irrational number, the terms with π cannot be combined with the integer terms. The final answer in standard form is: 4x^2 + 28x + 49 - πx^2 + 4πx - 4π, or equivalently 4x^2 + 28x + 49 - π(x^2 - 4x + 4).

2. (3x² - 2x + 5)(2x - 3) = ? Answer: 6x³ - 13x² + 16x - 15 Solution: We have: (3x² - 2x + 5)(2x - 3) We will use the distributive property (also called FOIL for binomials, but here the first factor is a trinomial).
   Full step-by-step solution

   Let's multiply the two expressions step by step. We have: (3x² - 2x + 5)(2x - 3) We will use the distributive property (also called FOIL for binomials, but here the first factor is a trinomial). --- **Step 1: Distribute 2x to each term in the trinomial** 2x * 3x² = 6x³ 2x * (-2x) = -4x² 2x * 5 = 10x So from 2x we get: 6x³ - 4x² + 10x --- **Step 2: Distribute -3 to each term in the trinomial** -3 * 3x² = -9x² -3 * (-2x) = +6x -3 * 5 = -15 So from -3 we get: -9x² + 6x - 15 --- **Step 3: Add the results from Step 1 and Step 2** From Step 1: 6x³ - 4x² + 10x From Step 2: - 9x² + 6x - 15 Now combine like terms: - x³ terms: 6x³ - x² terms: -4x² - 9x² = -13x² - x terms: 10x + 6x = 16x - constant terms: -15 So we have: 6x³ - 13x² + 16x - 15 --- **Final Answer:** 6x³ - 13x² + 16x - 15

3. (7x² - 9x + 11)(3x² + 5x - 13) = ? Answer: 21x⁴ + 8x³ - 157x² + 172x - 143 Solution: Multiply 7x² by each term in (3x² + 5x - 13): 7x² × 3x² = 21x⁴ 7x² × 5x = 35x³ 7x² × (-13) = -91x² Multiply -9x by each term in (3x² + 5x - 13): -9x × 3x² = -27x³ -9x × 5x = -45x² -9x × (-13) = 117x Multiply 11 by each term in (3x² + 5x - 13): 11 × 3x² = 33x² 11 × 5x = 55x 11 × (-13) = -143 21x⁴…
   Full step-by-step solution

   Step 1: Multiply 7x² by each term in (3x² + 5x - 13): 7x² × 3x² = 21x⁴ 7x² × 5x = 35x³ 7x² × (-13) = -91x² Step 2: Multiply -9x by each term in (3x² + 5x - 13): -9x × 3x² = -27x³ -9x × 5x = -45x² -9x × (-13) = 117x Step 3: Multiply 11 by each term in (3x² + 5x - 13): 11 × 3x² = 33x² 11 × 5x = 55x 11 × (-13) = -143 Step 4: Combine all terms: 21x⁴ + (35x³ - 27x³) + (-91x² - 45x² + 33x²) + (117x + 55x) - 143 Step 5: Simplify: 21x⁴ + 8x³ + (-103x²) + 172x - 143 The answer is 21x⁴ + 8x³ - 103x² + 172x - 143.

4. A company is designing a new smartphone with a rectangular screen. The screen's length is represented by (4x + 3) inches and its width is (3x - 2) inches. To calculate the screen area for manufacturing purposes, the engineers need to multiply these binomial expressions. What polynomial in standard form represents the area of the smartphone screen? Answer: 12x^2 + x - 6 Solution: Write the area expression: (4x + 3)(3x - 2) Apply distributive property: 4x(3x - 2) + 3(3x - 2) Multiply: 4x × 3x = 12x^2 Multiply: 4x × (-2) = -8x Multiply: 3 × 3x = 9x Multiply: 3 × (-2) = -6 Combine all terms: 12x^2 - 8x + 9x - 6 Combine like terms: 12x^2 + x - 6 The answer is 12x^2 + x - 6.
   Full step-by-step solution

   Step 1: Write the area expression: (4x + 3)(3x - 2) Step 2: Apply distributive property: 4x(3x - 2) + 3(3x - 2) Step 3: Multiply: 4x × 3x = 12x^2 Step 4: Multiply: 4x × (-2) = -8x Step 5: Multiply: 3 × 3x = 9x Step 6: Multiply: 3 × (-2) = -6 Step 7: Combine all terms: 12x^2 - 8x + 9x - 6 Step 8: Combine like terms: 12x^2 + x - 6 The answer is 12x^2 + x - 6.

5. Mason is designing a square patio that has a side length of (4x + 7) feet. Inside the patio, there is a square garden bed in one corner with side length (x + 2) feet. The rest of the patio is paved with tiles. Using polynomial multiplication, find the area of the paved region in expanded standard form. Answer: 15x^2 + 52x + 45 Solution: Find the area of the entire patio. Area = side × side = (4x + 7)(4x + 7). Use FOIL: (4x)(4x) = 16x^2, (4x)(7) = 28x, (7)(4x) = 28x, (7)(7) = 49.
   Full step-by-step solution

   Step 1: Find the area of the entire patio. Area = side × side = (4x + 7)(4x + 7). Use FOIL: (4x)(4x) = 16x^2, (4x)(7) = 28x, (7)(4x) = 28x, (7)(7) = 49. Combine like terms: 16x^2 + (28x + 28x) + 49 = 16x^2 + 56x + 49. Step 2: Find the area of the garden bed. Area = side × side = (x + 2)(x + 2). Use FOIL: (x)(x) = x^2, (x)(2) = 2x, (2)(x) = 2x, (2)(2) = 4. Combine like terms: x^2 + (2x + 2x) + 4 = x^2 + 4x + 4. Step 3: Subtract the garden bed area from the patio area to find the paved area: (16x^2 + 56x + 49) - (x^2 + 4x + 4). Subtract term by term: 16x^2 - x^2 = 15x^2, 56x - 4x = 52x, 49 - 4 = 45. Step 4: The paved area is 15x^2 + 52x + 45 in standard form. The answer is 15x^2 + 52x + 45.

6. Aroha is designing a rectangular stained glass window. The window's dimensions are expressed as (4x + 9) inches by (3x - 7) inches. Inside the window, there is a square panel of clear glass with side length (x + 2) inches, positioned in one corner of the rectangle. Using polynomial multiplication and the distributive property, find the area of the colored glass surrounding the clear square panel in expanded polynomial form. Answer: 11x^2 + 13x - 67 Solution: Find the area of the whole rectangular window. Area = length × width = (4x + 9)(3x - 7) (4x)(3x) = 12x^2 (4x)(-7) = -28x (9)(3x) = 27x (9)(-7) = -63 Combine like terms: 12x^2 + (-28x + 27x) - 63 = 12x^2 - x - 63 Find the area of the clear square panel.
   Full step-by-step solution

   Step 1: Find the area of the whole rectangular window. Area = length × width = (4x + 9)(3x - 7) Using distributive property (FOIL): (4x)(3x) = 12x^2 (4x)(-7) = -28x (9)(3x) = 27x (9)(-7) = -63 Combine like terms: 12x^2 + (-28x + 27x) - 63 = 12x^2 - x - 63 Step 2: Find the area of the clear square panel. Side = (x + 2) inches Area = (x + 2)^2 = (x + 2)(x + 2) Using distributive property: (x)(x) = x^2 (x)(2) = 2x (2)(x) = 2x (2)(2) = 4 Combine like terms: x^2 + (2x + 2x) + 4 = x^2 + 4x + 4 Step 3: Subtract the square's area from the rectangle's area to find the colored glass area. Colored area = (12x^2 - x - 63) - (x^2 + 4x + 4) Distribute the negative sign: 12x^2 - x - 63 - x^2 - 4x - 4 Combine like terms: 12x^2 - x^2 = 11x^2 -x - 4x = -5x -63 - 4 = -67 So, colored area = 11x^2 - 5x - 67 Step 4: Check if the problem intended the square in one corner. The description says "positioned in one corner," so the subtraction is correct. The final answer is 11x^2 - 5x - 67.

7. (3x² - 5x + 7)(x² + 9x - 11) = ? Answer: 3x⁴ + 22x³ - 65x² + 118x - 77 Solution: Multiply 3x² by each term in (x² + 9x - 11): 3x² × x² = 3x⁴ 3x² × 9x = 27x³ 3x² × (-11) = -33x² Multiply -5x by each term in (x² + 9x - 11): -5x × x² = -5x³ -5x × 9x = -45x² -5x × (-11) = 55x Multiply 7 by each term in (x² + 9x - 11): 7 × x² = 7x² 7 × 9x = 63x 7 × (-11) = -77 3x⁴ + (27x³ - 5x³)…
   Full step-by-step solution

   Step 1: Multiply 3x² by each term in (x² + 9x - 11): 3x² × x² = 3x⁴ 3x² × 9x = 27x³ 3x² × (-11) = -33x² Step 2: Multiply -5x by each term in (x² + 9x - 11): -5x × x² = -5x³ -5x × 9x = -45x² -5x × (-11) = 55x Step 3: Multiply 7 by each term in (x² + 9x - 11): 7 × x² = 7x² 7 × 9x = 63x 7 × (-11) = -77 Step 4: Combine all terms: 3x⁴ + (27x³ - 5x³) + (-33x² - 45x² + 7x²) + (55x + 63x) - 77 Step 5: Simplify: 3x⁴ + 22x³ + (-71x²) + 118x - 77 The answer is 3x⁴ + 22x³ - 71x² + 118x - 77.