Polynomial Division

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Answer Key & Explanations

Polynomial Division · Grade 9 · Worksheet 1

1. (45x⁸y⁷ - 60x⁶y⁹ + 75x⁷y⁵) ÷ (15x⁴y³) = ? Answer: 3x⁴y⁴ - 4x²y⁶ + 5x³y² Solution: Write the division as separate fractions: (45x⁸y⁷)/(15x⁴y³) - (60x⁶y⁹)/(15x⁴y³) + (75x⁷y⁵)/(15x⁴y³) Simplify the first term: 45/15 = 3, x⁸/x⁴ = x⁴, y⁷/y³ = y⁴ → 3x⁴y⁴ Simplify the second term: 60/15 = 4, x⁶/x⁴ = x², y⁹/y³ = y⁶ → 4x²y⁶ Simplify the third term: 75/15 = 5, x⁷/x⁴ = x³, y⁵/y³ = y² →…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (45x⁸y⁷)/(15x⁴y³) - (60x⁶y⁹)/(15x⁴y³) + (75x⁷y⁵)/(15x⁴y³) Step 2: Simplify the first term: 45/15 = 3, x⁸/x⁴ = x⁴, y⁷/y³ = y⁴ → 3x⁴y⁴ Step 3: Simplify the second term: 60/15 = 4, x⁶/x⁴ = x², y⁹/y³ = y⁶ → 4x²y⁶ Step 4: Simplify the third term: 75/15 = 5, x⁷/x⁴ = x³, y⁵/y³ = y² → 5x³y² Step 5: Combine the simplified terms: 3x⁴y⁴ - 4x²y⁶ + 5x³y² The answer is 3x⁴y⁴ - 4x²y⁶ + 5x³y².

2. A triangular garden is designed with vertices at coordinates A(0,0), B(4x,0), and C(0,3x). The gardener wants to divide the triangle into smaller congruent triangles by drawing lines parallel to the sides, creating a grid pattern. If the area of the original triangle is represented by the polynomial (6x²) square units and each small triangle has area (x²/2) square units, how many small congruent triangles are created in total? Answer: 12 Solution: The area of the original triangle is given as 6x² square units. The area of each small congruent triangle is given as x²/2 square units.
   Full step-by-step solution

   Step 1: The area of the original triangle is given as 6x² square units. Step 2: The area of each small congruent triangle is given as x²/2 square units. Step 3: To find the number of small triangles, divide the total area by the area of one small triangle: (6x²) ÷ (x²/2) Step 4: Dividing by a fraction is the same as multiplying by its reciprocal: 6x² × (2/x²) Step 5: Simplify the expression: 6 × 2 × (x²/x²) = 12 × 1 = 12 Step 6: Therefore, there are 12 small congruent triangles in total. The answer is 12.

3. (24x⁷y⁵ - 36x⁵y⁷ + 48x⁶y⁴) ÷ (12x³y²) = ? Answer: 2x⁴y³ - 3x²y⁵ + 4x³y² Solution: Write the division as separate fractions: (24x⁷y⁵)/(12x³y²) - (36x⁵y⁷)/(12x³y²) + (48x⁶y⁴)/(12x³y²) Simplify the first term: 24/12 = 2, x⁷/x³ = x⁴, y⁵/y² = y³ → 2x⁴y³ Simplify the second term: 36/12 = 3, x⁵/x³ = x², y⁷/y² = y⁵ → 3x²y⁵ Simplify the third term: 48/12 = 4, x⁶/x³ = x³, y⁴/y² = y² →…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (24x⁷y⁵)/(12x³y²) - (36x⁵y⁷)/(12x³y²) + (48x⁶y⁴)/(12x³y²) Step 2: Simplify the first term: 24/12 = 2, x⁷/x³ = x⁴, y⁵/y² = y³ → 2x⁴y³ Step 3: Simplify the second term: 36/12 = 3, x⁵/x³ = x², y⁷/y² = y⁵ → 3x²y⁵ Step 4: Simplify the third term: 48/12 = 4, x⁶/x³ = x³, y⁴/y² = y² → 4x³y² Step 5: Combine the simplified terms: 2x⁴y³ - 3x²y⁵ + 4x³y² The answer is 2x⁴y³ - 3x²y⁵ + 4x³y².

4. A rectangular garden is designed with length (3x² + 6x) meters and width (2x) meters. The garden is divided into equal square sections by drawing vertical lines. If each square section has area 2x² square meters, how many square sections are created? Answer: 3x + 6 Solution: When dividing a larger area into smaller equal sections, the number of sections can be found by dividing the total area by the area of one section.
   Full step-by-step solution

   When dividing a larger area into smaller equal sections, the number of sections can be found by dividing the total area by the area of one section. This principle applies to geometric division problems where a shape is partitioned into congruent smaller shapes.

5. (18x⁵y³ - 27x⁴y⁴ + 36x³y²) ÷ (9x²y) = ? Answer: 2x³y² - 3x²y³ + 4xy Solution: Write the division as separate fractions: (18x⁵y³)/(9x²y) - (27x⁴y⁴)/(9x²y) + (36x³y²)/(9x²y) Divide coefficients: 18 ÷ 9 = 2, 27 ÷ 9 = 3, 36 ÷ 9 = 4 Apply exponent rules for x terms: x⁵ ÷ x² = x³, x⁴ ÷ x² = x², x³ ÷ x² = x¹ Apply exponent rules for y terms: y³ ÷ y¹ = y², y⁴ ÷ y¹ = y³, y² ÷ y¹ =…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (18x⁵y³)/(9x²y) - (27x⁴y⁴)/(9x²y) + (36x³y²)/(9x²y) Step 2: Divide coefficients: 18 ÷ 9 = 2, 27 ÷ 9 = 3, 36 ÷ 9 = 4 Step 3: Apply exponent rules for x terms: x⁵ ÷ x² = x³, x⁴ ÷ x² = x², x³ ÷ x² = x¹ Step 4: Apply exponent rules for y terms: y³ ÷ y¹ = y², y⁴ ÷ y¹ = y³, y² ÷ y¹ = y¹ Step 5: Combine results: 2x³y² - 3x²y³ + 4xy The answer is 2x³y² - 3x²y³ + 4xy.

6. (24x⁶y⁵ - 36x⁴y⁷ + 48x⁵y⁴) ÷ (4x²y²) = ? Answer: 6x⁴y³ - 9x²y⁵ + 12x³y² Solution: Write the division as separate fractions: (24x⁶y⁵)/(4x²y²) - (36x⁴y⁷)/(4x²y²) + (48x⁵y⁴)/(4x²y²) Simplify the first term: 24/4 = 6, x⁶/x² = x⁴, y⁵/y² = y³ → 6x⁴y³ Simplify the second term: 36/4 = 9, x⁴/x² = x², y⁷/y² = y⁵ → 9x²y⁵ Simplify the third term: 48/4 = 12, x⁵/x² = x³, y⁴/y² = y² →…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (24x⁶y⁵)/(4x²y²) - (36x⁴y⁷)/(4x²y²) + (48x⁵y⁴)/(4x²y²) Step 2: Simplify the first term: 24/4 = 6, x⁶/x² = x⁴, y⁵/y² = y³ → 6x⁴y³ Step 3: Simplify the second term: 36/4 = 9, x⁴/x² = x², y⁷/y² = y⁵ → 9x²y⁵ Step 4: Simplify the third term: 48/4 = 12, x⁵/x² = x³, y⁴/y² = y² → 12x³y² Step 5: Combine the simplified terms: 6x⁴y³ - 9x²y⁵ + 12x³y² The answer is 6x⁴y³ - 9x²y⁵ + 12x³y².

7. (36x⁶y⁵ - 81x⁴y⁷ + 126x⁵y⁶) ÷ (9x²y³) = ? Answer: 4x⁴y² - 9x²y⁴ + 14x³y³ Solution: Write the division as separate fractions: (36x⁶y⁵)/(9x²y³) - (81x⁴y⁷)/(9x²y³) + (126x⁵y⁶)/(9x²y³) 36/9 = 4 x⁶/x² = x⁴ y⁵/y³ = y² Result: 4x⁴y² 81/9 = 9 x⁴/x² = x² y⁷/y³ = y⁴ Result: 9x²y⁴ 126/9 = 14 x⁵/x² = x³ y⁶/y³ = y³ Result: 14x³y³ Combine the simplified terms with the original signs: 4x⁴y²…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (36x⁶y⁵)/(9x²y³) - (81x⁴y⁷)/(9x²y³) + (126x⁵y⁶)/(9x²y³) Step 2: Simplify the first term: 36/9 = 4 x⁶/x² = x⁴ y⁵/y³ = y² Result: 4x⁴y² Step 3: Simplify the second term: 81/9 = 9 x⁴/x² = x² y⁷/y³ = y⁴ Result: 9x²y⁴ Step 4: Simplify the third term: 126/9 = 14 x⁵/x² = x³ y⁶/y³ = y³ Result: 14x³y³ Step 5: Combine the simplified terms with the original signs: 4x⁴y² - 9x²y⁴ + 14x³y³ The answer is 4x⁴y² - 9x²y⁴ + 14x³y³.

8. (28a⁹b⁶ - 42a⁷b⁸ + 56a⁸b⁵) ÷ (7a⁴b³) = ? Answer: 4a⁵b³ - 6a³b⁵ + 8a⁴b² Solution: Write the division as separate fractions: (28a⁹b⁶)/(7a⁴b³) - (42a⁷b⁸)/(7a⁴b³) + (56a⁸b⁵)/(7a⁴b³) Simplify the first term: 28/7 = 4, a⁹/a⁴ = a⁵, b⁶/b³ = b³ → 4a⁵b³ Simplify the second term: 42/7 = 6, a⁷/a⁴ = a³, b⁸/b³ = b⁵ → 6a³b⁵ Simplify the third term: 56/7 = 8, a⁸/a⁴ = a⁴, b⁵/b³ = b² → 8a⁴b²…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (28a⁹b⁶)/(7a⁴b³) - (42a⁷b⁸)/(7a⁴b³) + (56a⁸b⁵)/(7a⁴b³) Step 2: Simplify the first term: 28/7 = 4, a⁹/a⁴ = a⁵, b⁶/b³ = b³ → 4a⁵b³ Step 3: Simplify the second term: 42/7 = 6, a⁷/a⁴ = a³, b⁸/b³ = b⁵ → 6a³b⁵ Step 4: Simplify the third term: 56/7 = 8, a⁸/a⁴ = a⁴, b⁵/b³ = b² → 8a⁴b² Step 5: Combine the simplified terms: 4a⁵b³ - 6a³b⁵ + 8a⁴b² The answer is 4a⁵b³ - 6a³b⁵ + 8a⁴b².