Polynomial Division

lessonbunny.com

Answer Key & Explanations

Polynomial Division · Grade 9 · Worksheet 2

1. A triangular garden plot is designed with vertices at coordinates A(0,0), B(4x,0), and C(0,3x) on a coordinate plane. The gardener wants to divide the triangle into equal-area strips by drawing lines parallel to the base AB. If each strip has a height of x units, what simplified polynomial expression represents the area of the third strip from the top? Answer: 5x²/2 Solution: The overall triangle has base = 4x and height = 3x, so total area = (1/2) × base × height = (1/2) × 4x × 3x = 6x² The strips are drawn parallel to the base AB, with each strip having height x.
   Full step-by-step solution

   Step 1: The overall triangle has base = 4x and height = 3x, so total area = (1/2) × base × height = (1/2) × 4x × 3x = 6x² Step 2: The strips are drawn parallel to the base AB, with each strip having height x. Since the total height is 3x, there are 3 strips total. Step 3: The third strip from the top is actually the bottom strip. To find its area, we can calculate the area of the large triangle and subtract the area of the smaller triangle above it. Step 4: The smaller triangle above the bottom strip has height 2x (since we're removing the bottom strip of height x from the total height of 3x). Step 5: Since the triangles are similar, the base of the smaller triangle is proportional to its height: base = (2x/3x) × 4x = (2/3) × 4x = 8x/3 Step 6: Area of smaller triangle = (1/2) × (8x/3) × 2x = (1/2) × (16x²/3) = 8x²/3 Step 7: Area of bottom strip = total area - area of smaller triangle = 6x² - 8x²/3 = 18x²/3 - 8x²/3 = 10x²/3 Step 8: Simplify 10x²/3 = (10/3)x² The answer is (10/3)x².

2. (18x⁵y⁴ - 27x⁴y³ + 36x³y²) ÷ (9x²y) = ? Answer: 2x³y³ - 3x²y² + 4xy Solution: Write the division as separate fractions: (18x⁵y⁴)/(9x²y) - (27x⁴y³)/(9x²y) + (36x³y²)/(9x²y) Simplify the first term: 18/9 = 2, x⁵/x² = x³, y⁴/y = y³ → 2x³y³ Simplify the second term: 27/9 = 3, x⁴/x² = x², y³/y = y² → 3x²y² Simplify the third term: 36/9 = 4, x³/x² = x, y²/y = y → 4xy Combine…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (18x⁵y⁴)/(9x²y) - (27x⁴y³)/(9x²y) + (36x³y²)/(9x²y) Step 2: Simplify the first term: 18/9 = 2, x⁵/x² = x³, y⁴/y = y³ → 2x³y³ Step 3: Simplify the second term: 27/9 = 3, x⁴/x² = x², y³/y = y² → 3x²y² Step 4: Simplify the third term: 36/9 = 4, x³/x² = x, y²/y = y → 4xy Step 5: Combine the simplified terms: 2x³y³ - 3x²y² + 4xy The answer is 2x³y³ - 3x²y² + 4xy.

3. (15x⁴y³ - 25x³y⁵ + 35x²y⁴) ÷ (5x²y²) = ? Answer: 3x²y - 5xy³ + 7y² Solution: Write the division as separate fractions: (15x⁴y³)/(5x²y²) - (25x³y⁵)/(5x²y²) + (35x²y⁴)/(5x²y²) Divide coefficients: 15 ÷ 5 = 3, 25 ÷ 5 = 5, 35 ÷ 5 = 7 Apply exponent rules for x terms: x⁴ ÷ x² = x², x³ ÷ x² = x¹, x² ÷ x² = x⁰ = 1 Apply exponent rules for y terms: y³ ÷ y² = y¹, y⁵ ÷ y² = y³, y⁴…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (15x⁴y³)/(5x²y²) - (25x³y⁵)/(5x²y²) + (35x²y⁴)/(5x²y²) Step 2: Divide coefficients: 15 ÷ 5 = 3, 25 ÷ 5 = 5, 35 ÷ 5 = 7 Step 3: Apply exponent rules for x terms: x⁴ ÷ x² = x², x³ ÷ x² = x¹, x² ÷ x² = x⁰ = 1 Step 4: Apply exponent rules for y terms: y³ ÷ y² = y¹, y⁵ ÷ y² = y³, y⁴ ÷ y² = y² Step 5: Combine results: 3x²y - 5xy³ + 7y² The answer is 3x²y - 5xy³ + 7y².

4. (12x³y² - 18x²y⁴ + 6xy³) ÷ (3xy) = ? Answer: 4x²y - 6xy³ + 2y² Solution: (12x³y² - 18x²y⁴ + 6xy³) ÷ (3xy) We can split the numerator into three terms, each divided by 3xy: = (12x³y²)/(3xy) - (18x²y⁴)/(3xy) + (6xy³)/(3xy) 12 ÷ 3 = 4 18 ÷ 3 = 6 6 ÷ 3 = 2 4 * (x³y²)/(xy) - 6 * (x²y⁴)/(xy) + 2 * (xy³)/(xy) x³ / x¹ = x^(3-1) = x² x² / x¹ = x^(2-1) = x¹ = x x¹ / x¹ =…
   Full step-by-step solution

   Let's solve step by step. We are dividing: (12x³y² - 18x²y⁴ + 6xy³) ÷ (3xy) --- **Step 1: Write the division as separate fractions** We can split the numerator into three terms, each divided by 3xy: = (12x³y²)/(3xy) - (18x²y⁴)/(3xy) + (6xy³)/(3xy) --- **Step 2: Simplify coefficients** 12 ÷ 3 = 4 18 ÷ 3 = 6 6 ÷ 3 = 2 So we have: 4 * (x³y²)/(xy) - 6 * (x²y⁴)/(xy) + 2 * (xy³)/(xy) --- **Step 3: Simplify x terms** For (x³y²)/(xy): x³ / x¹ = x^(3-1) = x² For (x²y⁴)/(xy): x² / x¹ = x^(2-1) = x¹ = x For (xy³)/(xy): x¹ / x¹ = x^(1-1) = x⁰ = 1 So x terms simplify to: Term 1: x² Term 2: x Term 3: 1 --- **Step 4: Simplify y terms** For (x³y²)/(xy): y² / y¹ = y^(2-1) = y¹ = y For (x²y⁴)/(xy): y⁴ / y¹ = y^(4-1) = y³ For (xy³)/(xy): y³ / y¹ = y^(3-1) = y² --- **Step 5: Combine results** Term 1: 4 * x² * y = 4x²y Term 2: -6 * x * y³ = -6xy³ Term 3: +2 * 1 * y² = 2y² So the result is: 4x²y - 6xy³ + 2y² --- **Final answer:** 4x²y - 6xy³ + 2y²

5. (15x⁴y³ - 25x³y² + 35x²y⁴) ÷ (5x²y) = ? Answer: 3x²y² - 5xy + 7y³ Solution: Write the division as separate fractions: (15x⁴y³)/(5x²y) - (25x³y²)/(5x²y) + (35x²y⁴)/(5x²y) Divide coefficients: 15÷5=3, 25÷5=5, 35÷5=7 Apply exponent rules for x terms: x⁴÷x²=x², x³÷x²=x¹, x²÷x²=x⁰=1 Apply exponent rules for y terms: y³÷y=y², y²÷y=y¹, y⁴÷y=y³ Combine results: 3x²y² - 5x¹y¹ +…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (15x⁴y³)/(5x²y) - (25x³y²)/(5x²y) + (35x²y⁴)/(5x²y) Step 2: Divide coefficients: 15÷5=3, 25÷5=5, 35÷5=7 Step 3: Apply exponent rules for x terms: x⁴÷x²=x², x³÷x²=x¹, x²÷x²=x⁰=1 Step 4: Apply exponent rules for y terms: y³÷y=y², y²÷y=y¹, y⁴÷y=y³ Step 5: Combine results: 3x²y² - 5x¹y¹ + 7x⁰y³ Step 6: Simplify: 3x²y² - 5xy + 7y³ The answer is 3x²y² - 5xy + 7y³.

6. 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.

7. 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

8. (18x⁵y³ - 27x⁴y² + 36x³y⁴) ÷ (9x²y) = ? Answer: 2x³y² - 3x²y + 4xy³ Solution: Write the division as separate fractions: (18x⁵y³)/(9x²y) - (27x⁴y²)/(9x²y) + (36x³y⁴)/(9x²y) Simplify the first term: 18/9 = 2, x⁵/x² = x³, y³/y = y² → 2x³y² Simplify the second term: 27/9 = 3, x⁴/x² = x², y²/y = y → 3x²y Simplify the third term: 36/9 = 4, x³/x² = x, y⁴/y = y³ → 4xy³ Combine…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (18x⁵y³)/(9x²y) - (27x⁴y²)/(9x²y) + (36x³y⁴)/(9x²y) Step 2: Simplify the first term: 18/9 = 2, x⁵/x² = x³, y³/y = y² → 2x³y² Step 3: Simplify the second term: 27/9 = 3, x⁴/x² = x², y²/y = y → 3x²y Step 4: Simplify the third term: 36/9 = 4, x³/x² = x, y⁴/y = y³ → 4xy³ Step 5: Combine the simplified terms: 2x³y² - 3x²y + 4xy³ The answer is 2x³y² - 3x²y + 4xy³.