Polynomial Multiplication

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Polynomial Multiplication · Grade 9 · Worksheet 1

1. Hana is designing a rectangular raised garden bed for her school's environmental project. The length of the garden bed is represented by (3x + 7) decimeters and the width is represented by (2x - 5) decimeters. To order the correct amount of soil, Hana needs to calculate the area of the garden bed. What polynomial expression in standard form represents the area of the garden bed in square decimeters? Answer: 6x^2 - x - 35 Solution: Write the area formula: Area = length × width. Substitute the expressions: Area = (3x + 7)(2x - 5).
   Full step-by-step solution

   Step 1: Write the area formula: Area = length × width. Step 2: Substitute the expressions: Area = (3x + 7)(2x - 5). Step 3: Apply the distributive property (FOIL method): - First: 3x × 2x = 6x^2 - Outer: 3x × (-5) = -15x - Inner: 7 × 2x = 14x - Last: 7 × (-5) = -35 Step 4: Combine all terms: 6x^2 + (-15x) + 14x + (-35). Step 5: Combine like terms for the x terms: -15x + 14x = -x. Step 6: Write the final polynomial: 6x^2 - x - 35. The area of the garden bed is 6x^2 - x - 35 square decimeters.

2. (4x² - 3x + 7)(2x + 5) = ? Answer: 8x³ + 14x² - x + 35 Solution: Multiply each term in the first polynomial by each term in the second polynomial: (4x²)(2x) = 8x³ (4x²)(5) = 20x² (-3x)(2x) = -6x² (-3x)(5) = -15x (7)(2x) = 14x (7)(5) = 35 Write all terms: 8x³ + 20x² - 6x² - 15x + 14x + 35 8x³ + (20x² - 6x²) = 8x³ + 14x² (-15x + 14x) = -x + 35 Final answer: 8x³…
   Full step-by-step solution

   Step 1: Multiply each term in the first polynomial by each term in the second polynomial: (4x²)(2x) = 8x³ (4x²)(5) = 20x² (-3x)(2x) = -6x² (-3x)(5) = -15x (7)(2x) = 14x (7)(5) = 35 Step 2: Write all terms: 8x³ + 20x² - 6x² - 15x + 14x + 35 Step 3: Combine like terms: 8x³ + (20x² - 6x²) = 8x³ + 14x² (-15x + 14x) = -x + 35 Step 4: Final answer: 8x³ + 14x² - x + 35

3. 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.

4. (9x² - 11x + 13)(x² + 4x - 7) = ? Answer: 9x⁴ + 25x³ - 102x² + 129x - 91 Solution: Multiply 9x² by each term in (x² + 4x - 7): 9x² × x² = 9x⁴ 9x² × 4x = 36x³ 9x² × (-7) = -63x² Multiply -11x by each term in (x² + 4x - 7): -11x × x² = -11x³ -11x × 4x = -44x² -11x × (-7) = 77x Multiply 13 by each term in (x² + 4x - 7): 13 × x² = 13x² 13 × 4x = 52x 13 × (-7) = -91 9x⁴ + (36x³ -…
   Full step-by-step solution

   Step 1: Multiply 9x² by each term in (x² + 4x - 7): 9x² × x² = 9x⁴ 9x² × 4x = 36x³ 9x² × (-7) = -63x² Step 2: Multiply -11x by each term in (x² + 4x - 7): -11x × x² = -11x³ -11x × 4x = -44x² -11x × (-7) = 77x Step 3: Multiply 13 by each term in (x² + 4x - 7): 13 × x² = 13x² 13 × 4x = 52x 13 × (-7) = -91 Step 4: Combine all terms: 9x⁴ + (36x³ - 11x³) + (-63x² - 44x² + 13x²) + (77x + 52x) - 91 Step 5: Simplify by combining like terms: 9x⁴ + 25x³ + (-94x²) + 129x - 91 Step 6: Final answer: 9x⁴ + 25x³ - 94x² + 129x - 91

5. (4x² - 6x + 8)(2x² + 4x - 6) = ? Answer: 8x⁴ + 4x³ - 28x² + 68x - 48 Solution: Multiply 4x² by each term in (2x² + 4x - 6): 4x² × 2x² = 8x⁴ 4x² × 4x = 16x³ 4x² × (-6) = -24x² Multiply -6x by each term in (2x² + 4x - 6): -6x × 2x² = -12x³ -6x × 4x = -24x² -6x × (-6) = 36x Multiply 8 by each term in (2x² + 4x - 6): 8 × 2x² = 16x² 8 × 4x = 32x 8 × (-6) = -48 8x⁴ + (16x³ -…
   Full step-by-step solution

   Step 1: Multiply 4x² by each term in (2x² + 4x - 6): 4x² × 2x² = 8x⁴ 4x² × 4x = 16x³ 4x² × (-6) = -24x² Step 2: Multiply -6x by each term in (2x² + 4x - 6): -6x × 2x² = -12x³ -6x × 4x = -24x² -6x × (-6) = 36x Step 3: Multiply 8 by each term in (2x² + 4x - 6): 8 × 2x² = 16x² 8 × 4x = 32x 8 × (-6) = -48 Step 4: Combine all terms: 8x⁴ + (16x³ - 12x³) + (-24x² - 24x² + 16x²) + (36x + 32x) - 48 Step 5: Simplify by combining like terms: 8x⁴ + 4x³ + (-32x²) + 68x - 48 Step 6: Final answer: 8x⁴ + 4x³ - 32x² + 68x - 48

6. (4x³ - 3x² + 2x - 5)(2x² + x - 3) = ? Answer: 8x⁵ - 2x⁴ - 19x³ + 16x² + 11x - 15 Solution: Multiply 4x³ by each term in (2x² + x - 3): 4x³ × 2x² = 8x⁵ 4x³ × x = 4x⁴ 4x³ × (-3) = -12x³ Multiply -3x² by each term in (2x² + x - 3): -3x² × 2x² = -6x⁴ -3x² × x = -3x³ -3x² × (-3) = 9x² Multiply 2x by each term in (2x² + x - 3): 2x × 2x² = 4x³ 2x × x = 2x² 2x × (-3) = -6x Multiply -5 by each…
   Full step-by-step solution

   Step 1: Multiply 4x³ by each term in (2x² + x - 3): 4x³ × 2x² = 8x⁵ 4x³ × x = 4x⁴ 4x³ × (-3) = -12x³ Step 2: Multiply -3x² by each term in (2x² + x - 3): -3x² × 2x² = -6x⁴ -3x² × x = -3x³ -3x² × (-3) = 9x² Step 3: Multiply 2x by each term in (2x² + x - 3): 2x × 2x² = 4x³ 2x × x = 2x² 2x × (-3) = -6x Step 4: Multiply -5 by each term in (2x² + x - 3): -5 × 2x² = -10x² -5 × x = -5x -5 × (-3) = 15 Step 5: Combine all terms: 8x⁵ + (4x⁴ - 6x⁴) + (-12x³ - 3x³ + 4x³) + (9x² + 2x² - 10x²) + (-6x - 5x) + 15 Step 6: Simplify by combining like terms: 8x⁵ - 2x⁴ - 11x³ + x² - 11x + 15 The answer is 8x⁵ - 2x⁴ - 11x³ + x² - 11x + 15.

7. A rectangular community garden plot has dimensions (4x + 3) feet by (2x - 1) feet. The garden committee needs to calculate the total area to determine how many vegetable seedlings to purchase for spring planting. What is the area of the garden plot expressed as a polynomial in standard form? Answer: 8x^2 + 2x - 3 Solution: Write the area formula: Area = length × width = (4x + 3)(2x - 1) Use the distributive property: Multiply 4x by both terms in (2x - 1): 4x × 2x = 8x^2 and 4x × (-1) = -4x Multiply 3 by both terms in (2x - 1): 3 × 2x = 6x and 3 × (-1) = -3 Combine all terms: 8x^2 + (-4x) + 6x + (-3) = 8x^2 + 2x -…
   Full step-by-step solution

   Step 1: Write the area formula: Area = length × width = (4x + 3)(2x - 1) Step 2: Use the distributive property: Multiply 4x by both terms in (2x - 1): 4x × 2x = 8x^2 and 4x × (-1) = -4x Step 3: Multiply 3 by both terms in (2x - 1): 3 × 2x = 6x and 3 × (-1) = -3 Step 4: Combine all terms: 8x^2 + (-4x) + 6x + (-3) = 8x^2 + 2x - 3 Step 5: The polynomial is already in standard form (descending order of exponents) The area of the garden plot is 8x^2 + 2x - 3 square feet.

8. A rectangular solar panel has dimensions (4x + 3) meters by (2x - 1) meters. The energy company needs to calculate the total area to determine the panel's power output capacity. What polynomial expression in standard form represents the area of the solar panel? Answer: 8x² + 2x - 3 Solution: To multiply polynomials using the distributive property, you multiply each term in the first polynomial by each term in the second polynomial.
   Full step-by-step solution

   To multiply polynomials using the distributive property, you multiply each term in the first polynomial by each term in the second polynomial. For example, if you had (a + b)(c + d), you would calculate a×c + a×d + b×c + b×d. After multiplying all terms, combine any like terms to write the polynomial in standard form with descending powers.