Polynomial Division

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Polynomial Division · Grade 9 · Worksheet 3

1. Kaia is a materials engineer designing a composite beam for a bridge. The total cross-sectional area of the beam is represented by the polynomial 27x⁵y³ - 45x⁴y⁴ + 18x³y⁵ square centimeters. The beam is composed of several identical rectangular layers stacked vertically, each with a thickness of 9x²y³ centimeters. What simplified polynomial expression represents the total width of all layers combined? Answer: 3x³ - 5x²y + 2xy² Solution: Write the division problem: (27x⁵y³ - 45x⁴y⁴ + 18x³y⁵) ÷ (9x²y³) Divide each term of the polynomial by the monomial separately.
   Full step-by-step solution

   Step 1: Write the division problem: (27x⁵y³ - 45x⁴y⁴ + 18x³y⁵) ÷ (9x²y³) Step 2: Divide each term of the polynomial by the monomial separately. First term: 27x⁵y³ ÷ 9x²y³ Divide coefficients: 27 ÷ 9 = 3 Subtract exponents of x: 5 - 2 = 3, so x³ Subtract exponents of y: 3 - 3 = 0, so y⁰ = 1 Result: 3x³ Second term: -45x⁴y⁴ ÷ 9x²y³ Divide coefficients: -45 ÷ 9 = -5 Subtract exponents of x: 4 - 2 = 2, so x² Subtract exponents of y: 4 - 3 = 1, so y¹ Result: -5x²y Third term: 18x³y⁵ ÷ 9x²y³ Divide coefficients: 18 ÷ 9 = 2 Subtract exponents of x: 3 - 2 = 1, so x¹ Subtract exponents of y: 5 - 3 = 2, so y² Result: 2xy² Step 3: Combine the results: 3x³ - 5x²y + 2xy² The total width of all layers combined is 3x³ - 5x²y + 2xy² centimeters.

2. A robotics team is designing a solar panel array with a total area represented by the polynomial 15x⁴y³ - 25x³y⁴ + 35x²y⁵ square meters. They need to divide this area equally among 5x²y² identical solar panels. What simplified polynomial expression represents the area of each individual solar panel? Answer: 3x²y - 5xy² + 7y³ Solution: Dividing polynomials by monomials involves distributing the division across each term of the polynomial. For each term, you divide the coefficients and subtract the exponents of like variables.
   Full step-by-step solution

   Dividing polynomials by monomials involves distributing the division across each term of the polynomial. For each term, you divide the coefficients and subtract the exponents of like variables. This process is similar to simplifying fractions where you reduce both the numerical coefficients and the variable parts separately.

3. Sophia is analyzing a rectangular prism with a square base. The side length of the square base is (3x) units, and the height of the prism is (2x² + 5x - 7) units. She wants to find the volume of the prism and then divide that volume by the area of the square base to find the height. However, she makes a mistake: she accidentally divides the volume by the side length of the base (3x) instead of the area. What simplified polynomial expression does she incorrectly get for the height? Answer: 2x² + 5x - 7 Solution: The area of the square base is (3x)(3x) = 9x². The volume of the prism is (area of base)(height) = (9x²)(2x² + 5x - 7) = 18x⁴ + 45x³ - 63x².
   Full step-by-step solution

   Step 1: The area of the square base is (3x)(3x) = 9x². Step 2: The volume of the prism is (area of base)(height) = (9x²)(2x² + 5x - 7) = 18x⁴ + 45x³ - 63x². Step 3: Sophia incorrectly divides the volume by the side length (3x) instead of the area (9x²). So she computes (18x⁴ + 45x³ - 63x²) ÷ (3x). Step 4: Divide each term of the polynomial by 3x: - 18x⁴ ÷ 3x = 6x³ - 45x³ ÷ 3x = 15x² - (-63x²) ÷ 3x = -21x Step 5: The incorrect height she gets is 6x³ + 15x² - 21x. The answer is 6x³ + 15x² - 21x.

4. Sophia is an architect designing a modern art museum with a rectangular main hall. The total floor area of the hall is represented by the polynomial 28x⁴y² + 21x³y³ - 35x²y⁴ square feet. She plans to divide the hall into equal-sized gallery sections, each having an area of 7x²y² square feet. What simplified polynomial expression represents the number of gallery sections Sophia can create? Answer: 4x² + 3xy - 5y² Solution: Write the division problem: (28x⁴y² + 21x³y³ - 35x²y⁴) ÷ (7x²y²) Divide each term of the polynomial by the monomial separately: First term: 28x⁴y² ÷ 7x²y² = (28 ÷ 7)(x⁴ ÷ x²)(y² ÷ y²) = 4x² Second term: 21x³y³ ÷ 7x²y² = (21 ÷ 7)(x³ ÷ x²)(y³ ÷ y²) = 3xy Third term: -35x²y⁴ ÷ 7x²y² = (-35 ÷ 7)(x²…
   Full step-by-step solution

   Step 1: Write the division problem: (28x⁴y² + 21x³y³ - 35x²y⁴) ÷ (7x²y²) Step 2: Divide each term of the polynomial by the monomial separately: First term: 28x⁴y² ÷ 7x²y² = (28 ÷ 7)(x⁴ ÷ x²)(y² ÷ y²) = 4x² Second term: 21x³y³ ÷ 7x²y² = (21 ÷ 7)(x³ ÷ x²)(y³ ÷ y²) = 3xy Third term: -35x²y⁴ ÷ 7x²y² = (-35 ÷ 7)(x² ÷ x²)(y⁴ ÷ y²) = -5y² Step 3: Combine the results: 4x² + 3xy - 5y² The final answer is 4x² + 3xy - 5y².

5. (18x⁵y⁴ - 24x³y⁶ + 30x²y⁷) ÷ (6x²y³) = ? Answer: 3x³y - 4xy³ + 5y⁴ Solution: Write the division as separate fractions: (18x⁵y⁴)/(6x²y³) - (24x³y⁶)/(6x²y³) + (30x²y⁷)/(6x²y³) Simplify coefficients: 18/6 = 3, 24/6 = 4, 30/6 = 5 Apply exponent rules for x terms: x⁵/x² = x³, x³/x² = x¹, x²/x² = x⁰ = 1 Apply exponent rules for y terms: y⁴/y³ = y¹, y⁶/y³ = y³, y⁷/y³ = y⁴…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (18x⁵y⁴)/(6x²y³) - (24x³y⁶)/(6x²y³) + (30x²y⁷)/(6x²y³) Step 2: Simplify coefficients: 18/6 = 3, 24/6 = 4, 30/6 = 5 Step 3: Apply exponent rules for x terms: x⁵/x² = x³, x³/x² = x¹, x²/x² = x⁰ = 1 Step 4: Apply exponent rules for y terms: y⁴/y³ = y¹, y⁶/y³ = y³, y⁷/y³ = y⁴ Step 5: Combine results: 3x³y - 4xy³ + 5y⁴ The answer is 3x³y - 4xy³ + 5y⁴.

6. Liam is designing a rectangular garden with an area represented by the polynomial expression 6x³y² + 9x²y⁴ - 12x⁴y. He wants to divide the garden into equal sections along its width, where each section has a width of 3x²y. What polynomial expression represents the length of each garden section? Answer: 2xy + 3y³ - 4x² Solution: We are given the area of the rectangular garden as: Area = 6x³y² + 9x²y⁴ - 12x⁴y Width = 3x²y We need to find the length of each section.
   Full step-by-step solution

   We are given the area of the rectangular garden as: Area = 6x³y² + 9x²y⁴ - 12x⁴y The width of each section is: Width = 3x²y We need to find the length of each section. Since Area = Length × Width, Length = Area ÷ Width So: Length = (6x³y² + 9x²y⁴ - 12x⁴y) ÷ (3x²y) --- **Step 1: Break the division into separate terms** Length = (6x³y²)/(3x²y) + (9x²y⁴)/(3x²y) - (12x⁴y)/(3x²y) --- **Step 2: Simplify each term separately** First term: (6x³y²)/(3x²y) - Divide coefficients: 6/3 = 2 - For x: x³ / x² = x^(3-2) = x¹ = x - For y: y² / y¹ = y^(2-1) = y¹ = y So first term = 2xy Second term: (9x²y⁴)/(3x²y) - Divide coefficients: 9/3 = 3 - For x: x² / x² = x^(2-2) = x⁰ = 1 (so no x) - For y: y⁴ / y¹ = y^(4-1) = y³ So second term = 3y³ Third term: (12x⁴y)/(3x²y) - Divide coefficients: 12/3 = 4 - For x: x⁴ / x² = x^(4-2) = x² - For y: y / y = 1 (so no y) So third term = 4x² But note the original third term in the area is -12x⁴y, so: - ( -12x⁴y ) / (3x²y) = -4x² --- **Step 3: Combine the terms** Length = 2xy + 3y³ - 4x² --- **Final Answer:** Length = 2xy + 3y³ - 4x²

7. (18x⁵y³ - 24x⁴y² + 30x³y⁴) ÷ (6x²y) = ? Answer: 3x³y² - 4x²y + 5xy³ Solution: Write the division as separate fractions: (18x⁵y³)/(6x²y) - (24x⁴y²)/(6x²y) + (30x³y⁴)/(6x²y) Simplify the coefficients: 18/6 = 3, 24/6 = 4, 30/6 = 5 Simplify the x terms using exponent rules: x⁵/x² = x³, x⁴/x² = x², x³/x² = x¹ Simplify the y terms using exponent rules: y³/y¹ = y², y²/y¹ = y¹,…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (18x⁵y³)/(6x²y) - (24x⁴y²)/(6x²y) + (30x³y⁴)/(6x²y) Step 2: Simplify the coefficients: 18/6 = 3, 24/6 = 4, 30/6 = 5 Step 3: Simplify the x terms using exponent rules: x⁵/x² = x³, x⁴/x² = x², x³/x² = x¹ Step 4: Simplify the y terms using exponent rules: y³/y¹ = y², y²/y¹ = y¹, y⁴/y¹ = y³ Step 5: Combine the results: 3x³y² - 4x²y + 5xy³ The answer is 3x³y² - 4x²y + 5xy³.