Polynomial Multiplication

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Answer Key & Explanations

Polynomial Multiplication · Grade 9 · Worksheet 1

1. (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.

2. (4x² - 3x + 7)(2x² + x - 5) = ? Answer: 8x⁴ - 2x³ - 13x² + 38x - 35 Solution: Multiply each term in (4x² - 3x + 7) by each term in (2x² + x - 5) 4x² × 2x² = 8x⁴ 4x² × x = 4x³ 4x² × (-5) = -20x² -3x × 2x² = -6x³ -3x × x = -3x² -3x × (-5) = 15x 7 × 2x² = 14x² 7 × x = 7x 7 × (-5) = -35 Combine like terms: 8x⁴ + (4x³ - 6x³) + (-20x² - 3x² + 14x²) + (15x + 7x) - 35 Simplify:…
   Full step-by-step solution

   Step 1: Multiply each term in (4x² - 3x + 7) by each term in (2x² + x - 5) Step 2: 4x² × 2x² = 8x⁴ Step 3: 4x² × x = 4x³ Step 4: 4x² × (-5) = -20x² Step 5: -3x × 2x² = -6x³ Step 6: -3x × x = -3x² Step 7: -3x × (-5) = 15x Step 8: 7 × 2x² = 14x² Step 9: 7 × x = 7x Step 10: 7 × (-5) = -35 Step 11: Combine like terms: 8x⁴ + (4x³ - 6x³) + (-20x² - 3x² + 14x²) + (15x + 7x) - 35 Step 12: Simplify: 8x⁴ - 2x³ - 9x² + 22x - 35 Step 13: Correction: -20x² - 3x² + 14x² = -9x², and 15x + 7x = 22x Step 14: Final answer: 8x⁴ - 2x³ - 9x² + 22x - 35

3. A rectangular garden has a length of (3x + 2) meters and a width of (2x - 1) meters. The gardener wants to install a decorative border around the entire perimeter. Write a simplified polynomial expression that represents the area of the garden in square meters. Answer: 6x^2 + x - 2 Solution: We have a rectangular garden. Length = (3x + 2) meters Width = (2x - 1) meters We are asked for the area, not the perimeter.
   Full step-by-step solution

   Step 1: Understand the problem We have a rectangular garden. Length = (3x + 2) meters Width = (2x - 1) meters We are asked for the area, not the perimeter. Step 2: Recall the area formula for a rectangle Area = length × width Step 3: Substitute the given expressions Area = (3x + 2) × (2x - 1) Step 4: Expand using the distributive property (FOIL method) First: (3x)(2x) = 6x^2 Outer: (3x)(-1) = -3x Inner: (2)(2x) = 4x Last: (2)(-1) = -2 Step 5: Write out the full expression before simplifying Area = 6x^2 - 3x + 4x - 2 Step 6: Combine like terms The like terms are -3x and +4x. -3x + 4x = 1x (which is written as x) So: Area = 6x^2 + x - 2 Step 7: Final answer The simplified polynomial expression for the area is: 6x^2 + x - 2

4. (2x² - 3x + 4)(x² + 2x - 1) = ? Answer: 2x⁴ + x³ - 4x² + 11x - 4 Solution: Multiply 2x² by each term in (x² + 2x - 1): 2x² × x² = 2x⁴ 2x² × 2x = 4x³ 2x² × (-1) = -2x² Multiply -3x by each term in (x² + 2x - 1): -3x × x² = -3x³ -3x × 2x = -6x² -3x × (-1) = 3x Multiply 4 by each term in (x² + 2x - 1): 4 × x² = 4x² 4 × 2x = 8x 4 × (-1) = -4 2x⁴ + (4x³ - 3x³) + (-2x² - 6x²…
   Full step-by-step solution

   Step 1: Multiply 2x² by each term in (x² + 2x - 1): 2x² × x² = 2x⁴ 2x² × 2x = 4x³ 2x² × (-1) = -2x² Step 2: Multiply -3x by each term in (x² + 2x - 1): -3x × x² = -3x³ -3x × 2x = -6x² -3x × (-1) = 3x Step 3: Multiply 4 by each term in (x² + 2x - 1): 4 × x² = 4x² 4 × 2x = 8x 4 × (-1) = -4 Step 4: Combine all terms: 2x⁴ + (4x³ - 3x³) + (-2x² - 6x² + 4x²) + (3x + 8x) - 4 Step 5: Simplify by combining like terms: 2x⁴ + x³ + (-4x²) + 11x - 4 Final answer: 2x⁴ + x³ - 4x² + 11x - 4

5. A rectangular solar panel array is being designed for a school rooftop. The length of the array is represented by (4x + 3) feet and the width is (2x - 1) feet. The engineers need to calculate the total area to determine how many solar cells can be installed. What polynomial expression in standard form represents the area of the solar panel array? Answer: 8x^2 + 2x - 3 Solution: Write the area formula: Area = length × width Substitute the expressions: Area = (4x + 3)(2x - 1) First: 4x × 2x = 8x^2 Outer: 4x × (-1) = -4x Inner: 3 × 2x = 6x Last: 3 × (-1) = -3 Combine all terms: 8x^2 - 4x + 6x - 3 Combine like terms: 8x^2 + 2x - 3 The polynomial is already in standard form…
   Full step-by-step solution

   Step 1: Write the area formula: Area = length × width Step 2: Substitute the expressions: Area = (4x + 3)(2x - 1) Step 3: Use the distributive property (FOIL method): First: 4x × 2x = 8x^2 Outer: 4x × (-1) = -4x Inner: 3 × 2x = 6x Last: 3 × (-1) = -3 Step 4: Combine all terms: 8x^2 - 4x + 6x - 3 Step 5: Combine like terms: 8x^2 + 2x - 3 Step 6: The polynomial is already in standard form (descending powers of x) The area is represented by 8x^2 + 2x - 3

6. A rectangular garden has a length of (3x + 2) meters and a width of (2x - 1) meters. The gardener wants to install a decorative border around the entire garden. Write a simplified polynomial expression that represents the area of the garden in square meters. Answer: 6x^2 + x - 2 Solution: To find the area of a rectangle, we use the formula: Area = length × width. Length = (3x + 2) meters Width = (2x − 1) meters Write the expression for the area.
   Full step-by-step solution

   To find the area of a rectangle, we use the formula: Area = length × width. Given: Length = (3x + 2) meters Width = (2x − 1) meters Step 1: Write the expression for the area. Area = (3x + 2) × (2x − 1) Step 2: Expand using the distributive property (often called FOIL for binomials). First: Multiply the first terms: 3x × 2x = 6x² Outer: Multiply the outer terms: 3x × (−1) = −3x Inner: Multiply the inner terms: 2 × 2x = 4x Last: Multiply the last terms: 2 × (−1) = −2 Step 3: Write out all the terms from the expansion. Area = 6x² − 3x + 4x − 2 Step 4: Combine like terms (−3x + 4x). −3x + 4x = 1x, which is just x. Step 5: Write the simplified polynomial. Area = 6x² + x − 2 Final Answer: 6x² + x − 2

7. A rectangular garden has a length of (2x + 3) meters and a width of (x - 1) meters. If the garden is surrounded by a path of uniform width that extends 1 meter beyond the garden on all sides, what is the total area of the garden and path combined? Express your answer as a polynomial in standard form. Answer: 2x² + 9x + 5 Solution: When adding a border of uniform width around a rectangle, you add twice the border width to both the length and width.
   Full step-by-step solution

   When adding a border of uniform width around a rectangle, you add twice the border width to both the length and width. After finding the new dimensions, multiply them using the distributive property to expand the polynomial expression. This demonstrates how geometric area problems can be modeled using polynomial multiplication.

8. A rectangular solar panel array is being designed for a science center. The length of the array is represented by (4x + 3) meters and the width by (3x - 2) meters. The engineers need to calculate the total area to determine how many solar cells can be installed. What polynomial expression in standard form represents the area of the solar panel array? Answer: 12x^2 + x - 6 Solution: Write the area formula: Area = length × width Substitute the expressions: Area = (4x + 3)(3x - 2) First: 4x × 3x = 12x^2 Outer: 4x × (-2) = -8x Inner: 3 × 3x = 9x Last: 3 × (-2) = -6 Combine all terms: 12x^2 - 8x + 9x - 6 Combine like terms: 12x^2 + x - 6 The polynomial is already in standard…
   Full step-by-step solution

   Step 1: Write the area formula: Area = length × width Step 2: Substitute the expressions: Area = (4x + 3)(3x - 2) Step 3: Use distributive property (FOIL method): First: 4x × 3x = 12x^2 Outer: 4x × (-2) = -8x Inner: 3 × 3x = 9x Last: 3 × (-2) = -6 Step 4: Combine all terms: 12x^2 - 8x + 9x - 6 Step 5: Combine like terms: 12x^2 + x - 6 Step 6: The polynomial is already in standard form (descending powers of x) The area is represented by 12x^2 + x - 6 square meters.