Polynomial Multiplication

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Answer Key & Explanations

Polynomial Multiplication · Grade 9 · Worksheet 3

1. (5x² - 7x + 9)(3x² + 11x - 15) = ? Answer: 15x⁴ + 34x³ - 167x² + 204x - 135 Solution: Multiply 5x² by each term in (3x² + 11x - 15): 5x² × 3x² = 15x⁴ 5x² × 11x = 55x³ 5x² × (-15) = -75x² Multiply -7x by each term in (3x² + 11x - 15): -7x × 3x² = -21x³ -7x × 11x = -77x² -7x × (-15) = 105x Multiply 9 by each term in (3x² + 11x - 15): 9 × 3x² = 27x² 9 × 11x = 99x 9 × (-15) = -135…
   Full step-by-step solution

   Step 1: Multiply 5x² by each term in (3x² + 11x - 15): 5x² × 3x² = 15x⁴ 5x² × 11x = 55x³ 5x² × (-15) = -75x² Step 2: Multiply -7x by each term in (3x² + 11x - 15): -7x × 3x² = -21x³ -7x × 11x = -77x² -7x × (-15) = 105x Step 3: Multiply 9 by each term in (3x² + 11x - 15): 9 × 3x² = 27x² 9 × 11x = 99x 9 × (-15) = -135 Step 4: Combine all terms: 15x⁴ + (55x³ - 21x³) + (-75x² - 77x² + 27x²) + (105x + 99x) - 135 Step 5: Simplify: 15x⁴ + 34x³ + (-125x²) + 204x - 135 The answer is 15x⁴ + 34x³ - 125x² + 204x - 135.

2. Mere is designing a rectangular mural for the school hall. The mural's length is represented by (4x + 11) units and its width by (3x - 7) units. On a coordinate grid of the mural, a square section in the top-left corner, with side length (x + 2) units, will be painted in a special metallic color. What is the area of the remaining part of the mural (the part not painted metallic) expressed as a polynomial in standard form? Answer: 9x^2 + 7x - 81 Solution: Find the area of the entire rectangular mural. Area = length * width = (4x + 11)(3x - 7) (4x)(3x) = 12x^2 (4x)(-7) = -28x (11)(3x) = 33x (11)(-7) = -77 Combine like terms: 12x^2 + (-28x + 33x) - 77 = 12x^2 + 5x - 77 Total area = 12x^2 + 5x - 77 square units.
   Full step-by-step solution

   Step 1: Find the area of the entire rectangular mural. Area = length * width = (4x + 11)(3x - 7) Multiply using distributive property: (4x)(3x) = 12x^2 (4x)(-7) = -28x (11)(3x) = 33x (11)(-7) = -77 Combine like terms: 12x^2 + (-28x + 33x) - 77 = 12x^2 + 5x - 77 Total area = 12x^2 + 5x - 77 square units. Step 2: Find the area of the metallic square section. Area of square = side * side = (x + 2)(x + 2) Multiply using distributive property (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 Metallic area = x^2 + 4x + 4 square units. Step 3: Find the area of the remaining part. Remaining area = Total area - Metallic area Remaining area = (12x^2 + 5x - 77) - (x^2 + 4x + 4) Subtract term by term: 12x^2 - x^2 = 11x^2 5x - 4x = 1x or x -77 - 4 = -81 Remaining area = 11x^2 + x - 81 square units. The answer is 11x^2 + x - 81.

3. Hana is helping her family plan a new rectangular garden. The length of the garden is (3x + 11) meters, and the width is (2x - 7) meters. To buy the right amount of soil and fencing, Hana needs to find the area of the garden as a simplified polynomial expression. What is the area of the garden in square meters, written in standard form? Answer: 6x^2 - x - 77 Solution: Write the area formula for a rectangle: Area = length × width Substitute the given expressions: Area = (3x + 11)(2x - 7) First: 3x × 2x = 6x^2 Outer: 3x × (-7) = -21x Inner: 11 × 2x = 22x Last: 11 × (-7) = -77 Combine all the terms: 6x^2 + (-21x) + 22x + (-77) Combine like terms (-21x + 22x =…
   Full step-by-step solution

   Step 1: Write the area formula for a rectangle: Area = length × width Step 2: Substitute the given expressions: Area = (3x + 11)(2x - 7) Step 3: Apply the distributive property (FOIL method): First: 3x × 2x = 6x^2 Outer: 3x × (-7) = -21x Inner: 11 × 2x = 22x Last: 11 × (-7) = -77 Step 4: Combine all the terms: 6x^2 + (-21x) + 22x + (-77) Step 5: Combine like terms (-21x + 22x = x): 6x^2 + x - 77 The area of the garden in standard form is 6x^2 + x - 77 square meters.

4. (3x² - 7x + 11)(2x² + 5x - 9) = ? Answer: 6x⁴ + x³ - 38x² + 118x - 99 Solution: Multiply 3x² by each term in (2x² + 5x - 9): 3x² × 2x² = 6x⁴ 3x² × 5x = 15x³ 3x² × (-9) = -27x² Multiply -7x by each term in (2x² + 5x - 9): -7x × 2x² = -14x³ -7x × 5x = -35x² -7x × (-9) = 63x Multiply 11 by each term in (2x² + 5x - 9): 11 × 2x² = 22x² 11 × 5x = 55x 11 × (-9) = -99 6x⁴ + (15x³ -…
   Full step-by-step solution

   Step 1: Multiply 3x² by each term in (2x² + 5x - 9): 3x² × 2x² = 6x⁴ 3x² × 5x = 15x³ 3x² × (-9) = -27x² Step 2: Multiply -7x by each term in (2x² + 5x - 9): -7x × 2x² = -14x³ -7x × 5x = -35x² -7x × (-9) = 63x Step 3: Multiply 11 by each term in (2x² + 5x - 9): 11 × 2x² = 22x² 11 × 5x = 55x 11 × (-9) = -99 Step 4: Combine all terms: 6x⁴ + (15x³ - 14x³) + (-27x² - 35x² + 22x²) + (63x + 55x) - 99 Step 5: Simplify: 6x⁴ + x³ + (-40x²) + 118x - 99 The answer is 6x⁴ + x³ - 40x² + 118x - 99.

5. Noah is designing a rectangular courtyard for a community center. The length of the courtyard is represented by (4x + 7) meters and the width is represented by (3x - 2) meters. A circular fountain with a radius of (x + 1) meters will be placed in the center of the courtyard. Using polynomial multiplication, find the area of the courtyard that remains after subtracting the area of the circular fountain (use π = 3). Express your final answer as a polynomial in standard form. Answer: 12x² + 13x - 17 Solution: Find the area of the rectangular courtyard. Area of rectangle = length × width = (4x + 7)(3x - 2) (4x)(3x) = 12x² (4x)(-2) = -8x (7)(3x) = 21x (7)(-2) = -14 Combine like terms: 12x² + (-8x + 21x) - 14 = 12x² + 13x - 14 So, the area of the rectangle is 12x² + 13x - 14 square meters.
   Full step-by-step solution

   Step 1: Find the area of the rectangular courtyard. Area of rectangle = length × width = (4x + 7)(3x - 2) Use the distributive property (FOIL): (4x)(3x) = 12x² (4x)(-2) = -8x (7)(3x) = 21x (7)(-2) = -14 Combine like terms: 12x² + (-8x + 21x) - 14 = 12x² + 13x - 14 So, the area of the rectangle is 12x² + 13x - 14 square meters. Step 2: Find the area of the circular fountain. Area of circle = π × radius² Given π = 3 and radius = (x + 1) meters Area = 3 × (x + 1)² First, expand (x + 1)² = (x + 1)(x + 1) (x)(x) = x² (x)(1) = x (1)(x) = x (1)(1) = 1 Combine: x² + 2x + 1 Now multiply by 3: 3 × (x² + 2x + 1) = 3x² + 6x + 3 So, the area of the fountain is 3x² + 6x + 3 square meters. Step 3: Subtract the fountain area from the courtyard area. Remaining area = (12x² + 13x - 14) - (3x² + 6x + 3) Subtract term by term: 12x² - 3x² = 9x² 13x - 6x = 7x -14 - 3 = -17 Remaining area = 9x² + 7x - 17 Step 4: Verify the polynomial is in standard form (descending powers of x). The final answer is 9x² + 7x - 17. Answer: 9x² + 7x - 17

6. Mere is designing a rectangular garden for her community center. The length of the garden is represented by (3x + 11) meters and the width is represented by (2x - 5) meters. To order enough topsoil, Mere needs to calculate the area of the garden. What polynomial expression in standard form represents the area of the garden in square meters? Answer: 6x^2 + 7x - 55 Solution: Write the area formula: Area = length x width. Substitute the expressions: Area = (3x + 11)(2x - 5).
   Full step-by-step solution

   Step 1: Write the area formula: Area = length x width. Step 2: Substitute the expressions: Area = (3x + 11)(2x - 5). Step 3: Apply the distributive property (FOIL method): First: 3x * 2x = 6x^2 Outer: 3x * (-5) = -15x Inner: 11 * 2x = 22x Last: 11 * (-5) = -55 Step 4: Add all the terms: 6x^2 + (-15x) + 22x + (-55). Step 5: Combine like terms (-15x + 22x = 7x): 6x^2 + 7x - 55. Step 6: The polynomial in standard form is 6x^2 + 7x - 55. The answer is 6x^2 + 7x - 55.

7. (3x² - 5x + 7)(x² + 3x - 1) = ? Answer: 3x⁴ + 4x³ - 11x² + 26x - 7 Solution: Multiply 3x² by each term in (x² + 3x - 1): 3x² × x² = 3x⁴ 3x² × 3x = 9x³ 3x² × (-1) = -3x² Multiply -5x by each term in (x² + 3x - 1): -5x × x² = -5x³ -5x × 3x = -15x² -5x × (-1) = 5x Multiply 7 by each term in (x² + 3x - 1): 7 × x² = 7x² 7 × 3x = 21x 7 × (-1) = -7 3x⁴ + (9x³ - 5x³) + (-3x² -…
   Full step-by-step solution

   Step 1: Multiply 3x² by each term in (x² + 3x - 1): 3x² × x² = 3x⁴ 3x² × 3x = 9x³ 3x² × (-1) = -3x² Step 2: Multiply -5x by each term in (x² + 3x - 1): -5x × x² = -5x³ -5x × 3x = -15x² -5x × (-1) = 5x Step 3: Multiply 7 by each term in (x² + 3x - 1): 7 × x² = 7x² 7 × 3x = 21x 7 × (-1) = -7 Step 4: Combine all terms: 3x⁴ + (9x³ - 5x³) + (-3x² - 15x² + 7x²) + (5x + 21x) - 7 Step 5: Simplify by combining like terms: 3x⁴ + 4x³ + (-11x²) + 26x - 7 The answer is 3x⁴ + 4x³ - 11x² + 26x - 7.