Polynomial Multiplication

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Polynomial Multiplication · Grade 9 · Worksheet 3

1. A robotics team is designing a new solar panel with a rectangular active area. The length of the panel is represented by (4x + 3) centimeters and the width is (3x - 2) centimeters. To calculate the maximum power output, they need to determine the total area of the panel. What polynomial expression in standard form represents the area of the solar panel in square centimeters? 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 the 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 centimeters.

2. A tech company is designing a new smartphone with a rectangular screen. The screen's length is represented by (4x + 3) centimeters and its width by (2x - 1) centimeters. The engineers need to calculate the total screen area to determine the manufacturing cost. What polynomial expression represents the area of the screen in square centimeters? Answer: 8x^2 + 2x - 3 Solution: To multiply polynomials, we use the distributive property where each term in the first polynomial is multiplied by each term in the second polynomial. This is often called the FOIL method for binomials.
   Full step-by-step solution

   To multiply polynomials, we use the distributive property where each term in the first polynomial is multiplied by each term in the second polynomial. This is often called the FOIL method for binomials. The resulting terms are then combined by adding like terms to express the answer in standard polynomial form with descending exponents.

3. A rectangular solar panel has a length of (4x + 3) meters and a width of (2x - 1) meters. The manufacturer needs to calculate the total surface area to determine the energy output capacity. What polynomial expression represents the area of the solar panel in square meters? Answer: 8x^2 + 2x - 3 Solution: Area of a rectangle = length × width Area = (4x + 3)(2x - 1) Use distributive property: 4x(2x - 1) + 3(2x - 1) Multiply: 4x × 2x = 8x^2, 4x × (-1) = -4x, 3 × 2x = 6x, 3 × (-1) = -3 Combine terms: 8x^2 + (-4x + 6x) - 3 Simplify: 8x^2 + 2x - 3 The answer is 8x^2 + 2x - 3.
   Full step-by-step solution

   Step 1: Area of a rectangle = length × width Step 2: Area = (4x + 3)(2x - 1) Step 3: Use distributive property: 4x(2x - 1) + 3(2x - 1) Step 4: Multiply: 4x × 2x = 8x^2, 4x × (-1) = -4x, 3 × 2x = 6x, 3 × (-1) = -3 Step 5: Combine terms: 8x^2 + (-4x + 6x) - 3 Step 6: Simplify: 8x^2 + 2x - 3 The answer is 8x^2 + 2x - 3.

4. A community center is designing a new rectangular courtyard with a central fountain. The courtyard's length is represented by (4x + 3) meters and its width by (2x - 1) meters. The fountain occupies a square area in the center with sides of (x + 2) meters. What polynomial expression represents the remaining usable courtyard area around the fountain? Answer: 7x² - 5x - 7 Solution: When working with polynomial area problems, you multiply the expressions for length and width to find total area. If there's a region to exclude, you calculate that area separately and subtract it from the total.
   Full step-by-step solution

   When working with polynomial area problems, you multiply the expressions for length and width to find total area. If there's a region to exclude, you calculate that area separately and subtract it from the total. This demonstrates how polynomial multiplication and subtraction model real-world spatial planning scenarios.

5. (2x² + 3x - 4)(x² - 2x + 5) = ? Answer: 2x⁴ - x³ + 5x² + 23x - 20 Solution: Multiply 2x² by each term in (x² - 2x + 5): 2x² × x² = 2x⁴ 2x² × (-2x) = -4x³ 2x² × 5 = 10x² Multiply 3x by each term in (x² - 2x + 5): 3x × x² = 3x³ 3x × (-2x) = -6x² 3x × 5 = 15x Multiply -4 by each term in (x² - 2x + 5): -4 × x² = -4x² -4 × (-2x) = 8x -4 × 5 = -20 2x⁴ + (-4x³ + 3x³) + (10x² -…
   Full step-by-step solution

   Step 1: Multiply 2x² by each term in (x² - 2x + 5): 2x² × x² = 2x⁴ 2x² × (-2x) = -4x³ 2x² × 5 = 10x² Step 2: Multiply 3x by each term in (x² - 2x + 5): 3x × x² = 3x³ 3x × (-2x) = -6x² 3x × 5 = 15x Step 3: Multiply -4 by each term in (x² - 2x + 5): -4 × x² = -4x² -4 × (-2x) = 8x -4 × 5 = -20 Step 4: Combine all terms: 2x⁴ + (-4x³ + 3x³) + (10x² - 6x² - 4x²) + (15x + 8x) - 20 Step 5: Simplify by combining like terms: 2x⁴ - x³ + 0x² + 23x - 20 Step 6: Final answer: 2x⁴ - x³ + 23x - 20

6. (2x² - 7x + 3)(x² + 2x - 7) = ? Answer: 2x⁴ - 3x³ - 25x² + 55x - 21 Solution: Multiply 2x² by each term in (x² + 2x - 7): 2x² × x² = 2x⁴ 2x² × 2x = 4x³ 2x² × (-7) = -14x² Multiply -7x by each term in (x² + 2x - 7): -7x × x² = -7x³ -7x × 2x = -14x² -7x × (-7) = 49x Multiply 3 by each term in (x² + 2x - 7): 3 × x² = 3x² 3 × 2x = 6x 3 × (-7) = -21 2x⁴ + (4x³ - 7x³) + (-14x²…
   Full step-by-step solution

   Step 1: Multiply 2x² by each term in (x² + 2x - 7): 2x² × x² = 2x⁴ 2x² × 2x = 4x³ 2x² × (-7) = -14x² Step 2: Multiply -7x by each term in (x² + 2x - 7): -7x × x² = -7x³ -7x × 2x = -14x² -7x × (-7) = 49x Step 3: Multiply 3 by each term in (x² + 2x - 7): 3 × x² = 3x² 3 × 2x = 6x 3 × (-7) = -21 Step 4: Combine all terms: 2x⁴ + (4x³ - 7x³) + (-14x² - 14x² + 3x²) + (49x + 6x) - 21 Step 5: Simplify by combining like terms: 2x⁴ - 3x³ + (-25x²) + 55x - 21 Step 6: Final answer: 2x⁴ - 3x³ - 25x² + 55x - 21

7. A rectangular garden is designed with length (3x + 4) meters and width (2x - 1) meters. The garden is divided into four equal rectangular sections by two perpendicular pathways - one running the full length and one running the full width. What is the area of just one of the four garden sections in expanded polynomial form? Answer: 1.5x² + 2.5x - 1 Solution: Total area = length × width = (3x + 4)(2x - 1) (3x)(2x) = 6x² (3x)(-1) = -3x (4)(2x) = 8x (4)(-1) = -4 Total area = 6x² + (-3x + 8x) - 4 = 6x² + 5x - 4 Since the pathways divide the garden into 4 equal sections, each section gets 1/4 of the total area Area of one section = (6x² + 5x - 4) ÷ 4…
   Full step-by-step solution

   Step 1: Find the total area of the garden Total area = length × width = (3x + 4)(2x - 1) Step 2: Multiply the polynomials (3x)(2x) = 6x² (3x)(-1) = -3x (4)(2x) = 8x (4)(-1) = -4 Total area = 6x² + (-3x + 8x) - 4 = 6x² + 5x - 4 Step 3: Determine the area of one section Since the pathways divide the garden into 4 equal sections, each section gets 1/4 of the total area Area of one section = (6x² + 5x - 4) ÷ 4 Step 4: Divide each term by 4 6x² ÷ 4 = 1.5x² 5x ÷ 4 = 1.25x -4 ÷ 4 = -1 Area of one section = 1.5x² + 1.25x - 1 Step 5: Convert decimals to fractions for standard form 1.5 = 3/2 and 1.25 = 5/4 Area of one section = (3/2)x² + (5/4)x - 1 Step 6: Convert to decimal form for final answer Area of one section = 1.5x² + 1.25x - 1