Polynomial Multiplication

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Polynomial Multiplication · Grade 9 · Worksheet 2

1. A rectangular solar panel has dimensions (4x + 3) meters by (2x - 1) meters. The manufacturer needs to calculate the total area to determine the panel's energy output capacity. What polynomial expression in standard form represents the area of the solar panel in square meters? 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 area is 8x^2 + 2x - 3 square meters.
   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 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 The area is 8x^2 + 2x - 3 square meters.

2. A rectangular community garden plot has dimensions where the length is (4x + 3) feet and the width is (3x - 2) feet. The garden committee needs to calculate the total area to determine how many vegetable seedlings to purchase. What is the area of the garden plot expressed as a polynomial in standard form? 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 12x^2 + x - 6 square feet.

3. A technology 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. The engineers need to calculate the screen area to determine the optimal pixel density. What polynomial expression in standard form represents the area of the smartphone screen in square inches? Answer: 12x^2 + x - 6 Solution: Write the area formula: Area = length × width Substitute the expressions: Area = (4x + 3)(3x - 2) First terms: 4x × 3x = 12x^2 Outer terms: 4x × (-2) = -8x Inner terms: 3 × 3x = 9x Last terms: 3 × (-2) = -6 Combine all terms: 12x^2 - 8x + 9x - 6 Combine like terms: 12x^2 + (-8x + 9x) - 6 = 12x^2…
   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: Apply the distributive property (FOIL method): First terms: 4x × 3x = 12x^2 Outer terms: 4x × (-2) = -8x Inner terms: 3 × 3x = 9x Last terms: 3 × (-2) = -6 Step 4: Combine all terms: 12x^2 - 8x + 9x - 6 Step 5: Combine like terms: 12x^2 + (-8x + 9x) - 6 = 12x^2 + x - 6 Step 6: The polynomial is already in standard form with terms in descending order The final answer is 12x^2 + x - 6.

4. A rectangular garden has length (2x + 5) meters and width (x - 3) meters. Using the distributive property, find the polynomial expression that represents the area of the garden in expanded form. Answer: 2x^2 - x - 15 Solution: Recall the formula for the area of a rectangle. Area = length × width. Substitute the given expressions for length and width.
   Full step-by-step solution

   Step 1: Recall the formula for the area of a rectangle. Area = length × width. Step 2: Substitute the given expressions for length and width. Length = (2x + 5) meters Width = (x - 3) meters So, Area = (2x + 5)(x - 3) Step 3: Apply the distributive property (also called FOIL for binomials). Multiply each term in the first binomial by each term in the second binomial. First, multiply 2x by each term in (x - 3): 2x × x = 2x² 2x × (-3) = -6x Step 4: Multiply 5 by each term in (x - 3): 5 × x = 5x 5 × (-3) = -15 Step 5: Write down all the products from Steps 3 and 4: 2x² - 6x + 5x - 15 Step 6: Combine like terms (-6x and +5x): -6x + 5x = -1x, which is written as -x. Step 7: Write the final simplified polynomial: 2x² - x - 15 This is the expanded form of the area of the garden.

5. (7x² - 9x + 11)(3x² + 8x - 13) = ? Answer: 21x⁴ + 29x³ - 148x² + 205x - 143 Solution: Multiply 7x² by each term in (3x² + 8x - 13): 7x² × 3x² = 21x⁴ 7x² × 8x = 56x³ 7x² × (-13) = -91x² Multiply -9x by each term in (3x² + 8x - 13): -9x × 3x² = -27x³ -9x × 8x = -72x² -9x × (-13) = 117x Multiply 11 by each term in (3x² + 8x - 13): 11 × 3x² = 33x² 11 × 8x = 88x 11 × (-13) = -143 21x⁴…
   Full step-by-step solution

   Step 1: Multiply 7x² by each term in (3x² + 8x - 13): 7x² × 3x² = 21x⁴ 7x² × 8x = 56x³ 7x² × (-13) = -91x² Step 2: Multiply -9x by each term in (3x² + 8x - 13): -9x × 3x² = -27x³ -9x × 8x = -72x² -9x × (-13) = 117x Step 3: Multiply 11 by each term in (3x² + 8x - 13): 11 × 3x² = 33x² 11 × 8x = 88x 11 × (-13) = -143 Step 4: Combine all terms: 21x⁴ + (56x³ - 27x³) + (-91x² - 72x² + 33x²) + (117x + 88x) - 143 Step 5: Simplify: 21x⁴ + 29x³ + (-130x²) + 205x - 143 The answer is 21x⁴ + 29x³ - 130x² + 205x - 143.

6. A rectangular prism has dimensions (2x + 3) units by (x - 1) units by (x + 2) units. Using polynomial multiplication, find the volume of the prism expressed as a polynomial in standard form. Answer: 2x^3 + 5x^2 - x - 6 Solution: Write the volume formula for a rectangular prism: Volume = length × width × height Substitute the given dimensions: Volume = (2x + 3)(x - 1)(x + 2) First multiply (2x + 3)(x - 1): (2x)(x) = 2x^2 (2x)(-1) = -2x (3)(x) = 3x (3)(-1) = -3 Combine: 2x^2 + (-2x + 3x) - 3 = 2x^2 + x - 3 Now multiply…
   Full step-by-step solution

   Step 1: Write the volume formula for a rectangular prism: Volume = length × width × height Step 2: Substitute the given dimensions: Volume = (2x + 3)(x - 1)(x + 2) Step 3: First multiply (2x + 3)(x - 1): (2x)(x) = 2x^2 (2x)(-1) = -2x (3)(x) = 3x (3)(-1) = -3 Combine: 2x^2 + (-2x + 3x) - 3 = 2x^2 + x - 3 Step 4: Now multiply (2x^2 + x - 3)(x + 2): (2x^2)(x) = 2x^3 (2x^2)(2) = 4x^2 (x)(x) = x^2 (x)(2) = 2x (-3)(x) = -3x (-3)(2) = -6 Step 5: Combine like terms: 2x^3 + (4x^2 + x^2) + (2x - 3x) - 6 = 2x^3 + 5x^2 - x - 6 Step 6: The volume in standard form is 2x^3 + 5x^2 - x - 6

7. A tech company is designing a new smartphone with a rectangular screen. The screen's length is represented by (4x + 3) inches and its width is (2x - 1) inches. The engineers need to calculate the total display area to determine the pixel density requirements. What polynomial expression in standard form represents the area of the smartphone screen? 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 the polynomial 8x^2 + 2x - 3.

8. (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.