Polynomial Division

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Polynomial Division · Grade 9 · Worksheet 2

1. A robotics team is designing a solar-powered drone with wing panels that generate electricity. The total power output from all panels is represented by the polynomial 15x⁴y³ - 25x³y⁴ + 35x²y⁵ watts. If this power needs to be distributed equally among 5x²y² identical battery systems, what simplified polynomial expression represents the power allocated to each battery system? Answer: 3x²y - 5xy² + 7y³ Solution: Write the division problem: (15x⁴y³ - 25x³y⁴ + 35x²y⁵) ÷ (5x²y²) Divide each term of the polynomial by the monomial: First term: 15x⁴y³ ÷ 5x²y² = (15÷5)x^(4-2)y^(3-2) = 3x²y Second term: -25x³y⁴ ÷ 5x²y² = (-25÷5)x^(3-2)y^(4-2) = -5xy² Third term: 35x²y⁵ ÷ 5x²y² = (35÷5)x^(2-2)y^(5-2) = 7y³…
   Full step-by-step solution

   Step 1: Write the division problem: (15x⁴y³ - 25x³y⁴ + 35x²y⁵) ÷ (5x²y²) Step 2: Divide each term of the polynomial by the monomial: First term: 15x⁴y³ ÷ 5x²y² = (15÷5)x^(4-2)y^(3-2) = 3x²y Second term: -25x³y⁴ ÷ 5x²y² = (-25÷5)x^(3-2)y^(4-2) = -5xy² Third term: 35x²y⁵ ÷ 5x²y² = (35÷5)x^(2-2)y^(5-2) = 7y³ Step 3: Combine the results: 3x²y - 5xy² + 7y³ The answer is 3x²y - 5xy² + 7y³.

2. A triangular garden plot has vertices at coordinates A(0,0), B(4x,0), and C(2x,3x²). The gardener wants to divide the triangle into two equal areas by drawing a horizontal line parallel to the base AB. At what y-coordinate should this dividing line be placed? Answer: 3x²/√2 Solution: The original triangle has vertices A(0,0), B(4x,0), and C(2x,3x²). The base AB has length 4x, and the height is 3x². When we draw a horizontal line parallel to the base, it creates a smaller similar triangle at the top.
   Full step-by-step solution

   Step 1: The original triangle has vertices A(0,0), B(4x,0), and C(2x,3x²). The base AB has length 4x, and the height is 3x². Step 2: When we draw a horizontal line parallel to the base, it creates a smaller similar triangle at the top. Step 3: For the areas to be equal, the smaller triangle must have half the area of the original triangle. Step 4: The area ratio of similar triangles is the square of their linear scale factor. So if the area ratio is 1/2, then the linear scale factor is √(1/2) = 1/√2. Step 5: The height of the smaller triangle will be (1/√2) × 3x² = 3x²/√2. Step 6: Since the original height is 3x², the dividing line will be at y = 3x² - 3x²/√2 = 3x²(1 - 1/√2). Step 7: Wait, let me reconsider. The smaller triangle at the top has height h, and the linear scale factor is h/(3x²). The area ratio is [h/(3x²)]² = 1/2. Step 8: So [h/(3x²)]² = 1/2, which means h/(3x²) = 1/√2, so h = 3x²/√2. Step 9: Therefore, the dividing line should be placed at y = 3x²/√2. The answer is 3x²/√2.

3. A robotics team is designing a solar panel with a rectangular area represented by the polynomial 12x³y² + 18x²y³ - 6xy⁴ square centimeters. If they want to divide this panel into strips of equal width, where each strip has an area of 3xy² square centimeters, how many strips can they create? Express your answer as a simplified polynomial. Answer: 4x² + 6xy - 2y² Solution: Total area of the panel: 12x³y² + 18x²y³ - 6xy⁴ Area of each strip: 3xy² We want the number of strips = Total area ÷ Area per strip.
   Full step-by-step solution

   We are given: Total area of the panel: 12x³y² + 18x²y³ - 6xy⁴ Area of each strip: 3xy² We want the number of strips = Total area ÷ Area per strip. --- **Step 1: Write the division expression** Number of strips = (12x³y² + 18x²y³ - 6xy⁴) ÷ (3xy²) --- **Step 2: Break into separate fractions** (12x³y²)/(3xy²) + (18x²y³)/(3xy²) - (6xy⁴)/(3xy²) --- **Step 3: Simplify each term** First term: 12/3 = 4 x³/x = x² y²/y² = 1 So first term = 4x² Second term: 18/3 = 6 x²/x = x y³/y² = y So second term = 6xy Third term: 6/3 = 2 x/x = 1 y⁴/y² = y² So third term = 2y² (but note the original sign is minus) so it is - 2y² --- **Step 4: Combine results** 4x² + 6xy - 2y² --- **Final answer:** 4x² + 6xy - 2y²

4. Sophia is analyzing a geometric pattern formed by a sequence of rectangles drawn on a coordinate grid. The first rectangle has vertices at (0,0), (7a,0), (7a,2a²), and (0,2a²). The second rectangle is formed by increasing both the length and width by the same monomial factor. If the area of the second rectangle is represented by the polynomial (14a³ + 28a²) and its width is 7a, what simplified expression represents the length of the second rectangle? Answer: 2a + 4 Solution: The area of a rectangle is length times width. Here, area = (14a³ + 28a²) and width = 7a. To find the length, divide the area by the width: length = (14a³ + 28a²) / 7a.
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width. Here, area = (14a³ + 28a²) and width = 7a. Step 2: To find the length, divide the area by the width: length = (14a³ + 28a²) / 7a. Step 3: Divide each term of the polynomial by the monomial: (14a³ / 7a) + (28a² / 7a). Step 4: Simplify term by term: 14a³ / 7a = 2a², and 28a² / 7a = 4a. Step 5: So length = 2a² + 4a. Step 6: Factor out the common factor a: length = a(2a + 4). Step 7: Since the problem asks for the simplified expression representing the length, and the monomial factor a is part of the length, the fully simplified expression is 2a + 4 when considering the coefficient of a in the factorized form, but careful: Actually, length = 2a² + 4a, which is the simplest polynomial form. However, checking the problem: The first rectangle has length 2a², so the second rectangle's length should be larger. The correct simplified expression is 2a² + 4a. Step 8: Confirm by multiplying: (7a)(2a² + 4a) = 14a³ + 28a², which matches the given area. The answer is 2a² + 4a.

5. (15x⁴y³ - 25x³y² + 10x²y) ÷ (5x²y) = ? Answer: 3x²y² - 5xy + 2 Solution: Divide each term in the polynomial by the monomial (5x²y) First term: (15x⁴y³) ÷ (5x²y) = (15÷5)x^(4-2)y^(3-1) = 3x²y² Second term: (-25x³y²) ÷ (5x²y) = (-25÷5)x^(3-2)y^(2-1) = -5xy Third term: (10x²y) ÷ (5x²y) = (10÷5)x^(2-2)y^(1-1) = 2 Combine the results: 3x²y² - 5xy + 2 The answer is 3x²y² -…
   Full step-by-step solution

   Step 1: Divide each term in the polynomial by the monomial (5x²y) Step 2: First term: (15x⁴y³) ÷ (5x²y) = (15÷5)x^(4-2)y^(3-1) = 3x²y² Step 3: Second term: (-25x³y²) ÷ (5x²y) = (-25÷5)x^(3-2)y^(2-1) = -5xy Step 4: Third term: (10x²y) ÷ (5x²y) = (10÷5)x^(2-2)y^(1-1) = 2 Step 5: Combine the results: 3x²y² - 5xy + 2 The answer is 3x²y² - 5xy + 2.

6. A robotics team is designing a solar-powered drone with a wing surface area represented by the polynomial 15x⁴y³ - 25x³y⁴ + 10x²y⁵ square centimeters. They need to divide this wing area equally among 5x²y² identical solar cells. What simplified polynomial expression represents the area covered by each individual solar cell? Answer: 3x²y - 5xy² + 2y³ Solution: Write the division problem: (15x⁴y³ - 25x³y⁴ + 10x²y⁵) ÷ (5x²y²) Divide each term of the polynomial by the monomial separately: First term: 15x⁴y³ ÷ 5x²y² = (15 ÷ 5)(x⁴ ÷ x²)(y³ ÷ y²) = 3x²y Second term: -25x³y⁴ ÷ 5x²y² = (-25 ÷ 5)(x³ ÷ x²)(y⁴ ÷ y²) = -5xy² Third term: 10x²y⁵ ÷ 5x²y² = (10 ÷…
   Full step-by-step solution

   Step 1: Write the division problem: (15x⁴y³ - 25x³y⁴ + 10x²y⁵) ÷ (5x²y²) Step 2: Divide each term of the polynomial by the monomial separately: First term: 15x⁴y³ ÷ 5x²y² = (15 ÷ 5)(x⁴ ÷ x²)(y³ ÷ y²) = 3x²y Second term: -25x³y⁴ ÷ 5x²y² = (-25 ÷ 5)(x³ ÷ x²)(y⁴ ÷ y²) = -5xy² Third term: 10x²y⁵ ÷ 5x²y² = (10 ÷ 5)(x² ÷ x²)(y⁵ ÷ y²) = 2y³ Step 3: Combine the results: 3x²y - 5xy² + 2y³ The final answer is 3x²y - 5xy² + 2y³.

7. A rectangular garden is designed with length (3x² + 6x) meters and width (2x) meters. If the garden is divided into equal square plots by drawing lines parallel to the sides, creating a grid pattern, what is the area of each square plot when expressed in simplest polynomial form? Answer: 3x + 6 Solution: When dividing polynomials that represent dimensions of a rectangle, we're essentially finding the greatest common factor that divides both expressions evenly.
   Full step-by-step solution

   When dividing polynomials that represent dimensions of a rectangle, we're essentially finding the greatest common factor that divides both expressions evenly. This concept applies to many real-world scenarios like tiling floors, dividing land into plots, or creating uniform sections in manufacturing. The process involves identifying common factors in polynomial expressions, similar to how you might find the largest square that can evenly divide a rectangular area.