Polynomial Division

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Answer Key & Explanations

Polynomial Division · Grade 9 · Worksheet 1

1. Liam is designing a rectangular garden with an area represented by the polynomial expression 6x³ + 9x² - 12x square feet. He needs to divide the garden into three equal sections along its length, each with the same width. If the width of each section is 3x feet, what polynomial expression represents the length of one section? Answer: 2x² + 3x - 4 Solution: Dividing polynomials by monomials involves applying the distributive property of division over addition.
   Full step-by-step solution

   Dividing polynomials by monomials involves applying the distributive property of division over addition. Each term in the polynomial is divided by the monomial separately, which means dividing the coefficients and subtracting exponents of like variables. This process simplifies complex polynomial expressions into more manageable forms, which is particularly useful in geometric applications where dimensions are expressed algebraically.

2. A triangular garden plot is designed with vertices at coordinates (0,0), (4x,0), and (0,3x) on a coordinate plane. The gardener wants to divide the triangle into equal-area strips by drawing lines parallel to the y-axis, creating vertical sections. If the area of the triangle is represented by the polynomial 6x² and each strip has width x, what simplified expression represents the area of each strip? Answer: 3x/2 Solution: The triangle has vertices at (0,0), (4x,0), and (0,3x), forming a right triangle with base 4x and height 3x.
   Full step-by-step solution

   Step 1: The triangle has vertices at (0,0), (4x,0), and (0,3x), forming a right triangle with base 4x and height 3x. Step 2: The total area is given as 6x², which matches the formula for triangle area: (1/2) × base × height = (1/2) × 4x × 3x = 6x². Step 3: When divided into vertical strips of width x, there will be 4 strips total (since base = 4x and width of each strip = x). Step 4: Since the strips are parallel to the y-axis and the triangle's height decreases linearly from left to right, the areas of the strips are not equal. However, the problem states the gardener wants equal-area strips. Step 5: For equal-area strips, divide the total area by the number of strips: 6x² ÷ 4 = (6x²)/4 = (3x²)/2. Step 6: Simplify the expression: (3x²)/2 = 3x/2 × x, but since we want the area of each strip, the answer is 3x²/2. The area of each equal-area strip is 3x²/2.

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. Liam is designing a rectangular garden for his school's environmental club. The total area of the garden is represented by the polynomial expression 12x³y² + 18x²y⁵ - 6x⁴y. If the width of the garden is 6xy feet, what simplified algebraic expression represents the length of the garden? Answer: 2x²y + 3xy⁴ - x³ Solution: Area = 12x³y² + 18x²y⁵ − 6x⁴y Width = 6xy We want to find Length. Area = Length × Width So, Length = Area ÷ Width. Length = (12x³y² + 18x²y⁵ − 6x⁴y) ÷ (6xy) We can divide each term of the polynomial by 6xy separately.
   Full step-by-step solution

   We are given: Area = 12x³y² + 18x²y⁵ − 6x⁴y Width = 6xy We want to find Length. Step 1: Recall the formula for area of a rectangle: Area = Length × Width So, Length = Area ÷ Width. Step 2: Write the division: Length = (12x³y² + 18x²y⁵ − 6x⁴y) ÷ (6xy) Step 3: We can divide each term of the polynomial by 6xy separately. First term: 12x³y² ÷ (6xy) Divide coefficients: 12 ÷ 6 = 2 Divide x-powers: x³ ÷ x¹ = x^(3−1) = x² Divide y-powers: y² ÷ y¹ = y^(2−1) = y¹ = y So first term becomes: 2x²y Second term: 18x²y⁵ ÷ (6xy) Divide coefficients: 18 ÷ 6 = 3 Divide x-powers: x² ÷ x¹ = x^(2−1) = x¹ = x Divide y-powers: y⁵ ÷ y¹ = y^(5−1) = y⁴ So second term becomes: 3xy⁴ Third term: −6x⁴y ÷ (6xy) Divide coefficients: −6 ÷ 6 = −1 Divide x-powers: x⁴ ÷ x¹ = x^(4−1) = x³ Divide y-powers: y¹ ÷ y¹ = y^(1−1) = y⁰ = 1 (so no y remains) So third term becomes: −x³ Step 4: Combine the results: Length = 2x²y + 3xy⁴ − x³ Step 5: Check if terms can be reordered for standard form (usually highest power of x first): −x³ + 2x²y + 3xy⁴ But the given answer in the problem is written as: 2x²y + 3xy⁴ − x³, which is the same. Thus, the simplified algebraic expression for the length is: 2x²y + 3xy⁴ − x³

5. (27x⁷y⁵ - 72x⁵y⁷ + 45x⁶y⁴) ÷ (9x³y²) = ? Answer: 3x⁴y³ - 8x²y⁵ + 5x³y² Solution: Write the division as separate fractions: (27x⁷y⁵)/(9x³y²) - (72x⁵y⁷)/(9x³y²) + (45x⁶y⁴)/(9x³y²) Simplify the first term: 27/9 = 3, x⁷/x³ = x⁴, y⁵/y² = y³ → 3x⁴y³ Simplify the second term: 72/9 = 8, x⁵/x³ = x², y⁷/y² = y⁵ → 8x²y⁵ Simplify the third term: 45/9 = 5, x⁶/x³ = x³, y⁴/y² = y² → 5x³y²…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (27x⁷y⁵)/(9x³y²) - (72x⁵y⁷)/(9x³y²) + (45x⁶y⁴)/(9x³y²) Step 2: Simplify the first term: 27/9 = 3, x⁷/x³ = x⁴, y⁵/y² = y³ → 3x⁴y³ Step 3: Simplify the second term: 72/9 = 8, x⁵/x³ = x², y⁷/y² = y⁵ → 8x²y⁵ Step 4: Simplify the third term: 45/9 = 5, x⁶/x³ = x³, y⁴/y² = y² → 5x³y² Step 5: Combine the simplified terms: 3x⁴y³ - 8x²y⁵ + 5x³y² The answer is 3x⁴y³ - 8x²y⁵ + 5x³y².

6. (18x⁵y⁴ - 27x³y⁶ + 36x²y⁵) ÷ (9x²y³) = ? Answer: 2x³y - 3xy³ + 4y² Solution: Divide the first term: (18x⁵y⁴) ÷ (9x²y³) = (18/9)x^(5-2)y^(4-3) = 2x³y Divide the second term: (-27x³y⁶) ÷ (9x²y³) = (-27/9)x^(3-2)y^(6-3) = -3xy³ Divide the third term: (36x²y⁵) ÷ (9x²y³) = (36/9)x^(2-2)y^(5-3) = 4y² Combine the results: 2x³y - 3xy³ + 4y² The answer is 2x³y - 3xy³ + 4y².
   Full step-by-step solution

   Step 1: Divide the first term: (18x⁵y⁴) ÷ (9x²y³) = (18/9)x^(5-2)y^(4-3) = 2x³y Step 2: Divide the second term: (-27x³y⁶) ÷ (9x²y³) = (-27/9)x^(3-2)y^(6-3) = -3xy³ Step 3: Divide the third term: (36x²y⁵) ÷ (9x²y³) = (36/9)x^(2-2)y^(5-3) = 4y² Step 4: Combine the results: 2x³y - 3xy³ + 4y² The answer is 2x³y - 3xy³ + 4y².

7. A robotics team is designing a solar panel array with a total area represented by the polynomial 15x⁴y³ - 25x³y⁴ + 10x²y⁵ 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 + 2y³ Solution: Dividing polynomials by monomials involves applying the distributive property. Each term in the numerator is divided by the monomial in the denominator. For coefficients, you perform regular division.
   Full step-by-step solution

   Dividing polynomials by monomials involves applying the distributive property. Each term in the numerator is divided by the monomial in the denominator. For coefficients, you perform regular division. For variables, you subtract exponents according to the quotient rule of exponents. This process simplifies complex polynomial expressions into more manageable forms.

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.