Polynomial Division

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Polynomial Division · Grade 9 · Worksheet 3

1. A robotics team is designing a solar panel array with a total power output represented by the polynomial 24x⁴y³ - 18x³y⁴ + 12x²y⁵ watts. They need to distribute this power equally among 6x²y² identical solar cells. What simplified polynomial expression represents the power output of each individual solar cell? Answer: 4x²y - 3xy² + 2y³ Solution: Dividing polynomials by monomials involves applying the distributive property and using the quotient rule for exponents. When you divide terms with the same base, you subtract the exponents.
   Full step-by-step solution

   Dividing polynomials by monomials involves applying the distributive property and using the quotient rule for exponents. When you divide terms with the same base, you subtract the exponents. For example, if you had (8a³b² - 4a²b³) ÷ 2ab, you would divide each term separately and simplify the exponents.

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 solar panels that each have an area of 5x²y³ square meters. What simplified polynomial expression represents how many solar panels they can create? Answer: 3x² - 5xy + 7y² Solution: Dividing polynomials by monomials involves applying the distributive property of division over addition and subtraction.
   Full step-by-step solution

   Dividing polynomials by monomials involves applying the distributive property of division over addition and subtraction. Each term in the polynomial is divided by the monomial separately, which means dividing the coefficients and subtracting exponents of like variables according to the quotient rule of exponents. This process simplifies complex area division problems into manageable algebraic expressions.

3. (18x⁵y⁴ - 27x⁴y³ + 36x³y⁵) ÷ (9x²y²) = ? Answer: 2x³y² - 3x²y + 4xy³ Solution: Write the division as separate fractions: (18x⁵y⁴)/(9x²y²) - (27x⁴y³)/(9x²y²) + (36x³y⁵)/(9x²y²) Divide the coefficients: 18 ÷ 9 = 2, 27 ÷ 9 = 3, 36 ÷ 9 = 4 Apply the quotient rule for x exponents: x⁵ ÷ x² = x³, x⁴ ÷ x² = x², x³ ÷ x² = x¹ Apply the quotient rule for y exponents: y⁴ ÷ y² = y², y³…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (18x⁵y⁴)/(9x²y²) - (27x⁴y³)/(9x²y²) + (36x³y⁵)/(9x²y²) Step 2: Divide the coefficients: 18 ÷ 9 = 2, 27 ÷ 9 = 3, 36 ÷ 9 = 4 Step 3: Apply the quotient rule for x exponents: x⁵ ÷ x² = x³, x⁴ ÷ x² = x², x³ ÷ x² = x¹ Step 4: Apply the quotient rule for y exponents: y⁴ ÷ y² = y², y³ ÷ y² = y¹, y⁵ ÷ y² = y³ Step 5: Combine the results: 2x³y² - 3x²y + 4xy³ The answer is 2x³y² - 3x²y + 4xy³.

4. A robotics team is designing a solar panel array with a total power output represented by the polynomial 15x⁴y³ - 25x³y⁴ + 35x²y⁵ watts. They need to distribute this power equally among 5x²y² identical power converters. What simplified polynomial expression represents the power output that each converter must handle? Answer: 3x²y - 5xy² + 7y³ Solution: Dividing polynomials by monomials involves applying the quotient rule for exponents to each term. For each term in the polynomial, you divide the coefficients and subtract the exponents of matching variables.
   Full step-by-step solution

   Dividing polynomials by monomials involves applying the quotient rule for exponents to each term. For each term in the polynomial, you divide the coefficients and subtract the exponents of matching variables. This process is similar to simplifying fractions where both numerator and denominator contain variables with exponents.

5. (18x⁵y⁴ - 27x³y⁶ + 36x²y⁷) ÷ (9x²y³) = ? Answer: 2x³y - 3xy³ + 4y⁴ Solution: Write the division as separate fractions: (18x⁵y⁴)/(9x²y³) - (27x³y⁶)/(9x²y³) + (36x²y⁷)/(9x²y³) Divide coefficients: 18/9 = 2, 27/9 = 3, 36/9 = 4 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 Combine…
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (18x⁵y⁴)/(9x²y³) - (27x³y⁶)/(9x²y³) + (36x²y⁷)/(9x²y³) Step 2: Divide coefficients: 18/9 = 2, 27/9 = 3, 36/9 = 4 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: 2x³y - 3x¹y³ + 4x⁰y⁴ Step 6: Simplify: 2x³y - 3xy³ + 4y⁴ The answer is 2x³y - 3xy³ + 4y⁴.

6. Liam is designing a rectangular garden with an area represented by the polynomial 6x³ + 9x² - 12x square feet. He needs to divide the garden into three equal sections along its length using fencing that runs parallel to the width. If the width of each section is represented by 3x feet, what is the length of each section in terms of x? Answer: 2x² + 3x - 4 Solution: In polynomial division problems involving area, the total area is often represented as a polynomial, and when dividing by a monomial representing one dimension, you can find the other dimension by dividing each term of the polynomial separately.
   Full step-by-step solution

   In polynomial division problems involving area, the total area is often represented as a polynomial, and when dividing by a monomial representing one dimension, you can find the other dimension by dividing each term of the polynomial separately. This process demonstrates how algebraic expressions can model real-world spatial relationships, where dividing a complex area expression by a simpler width expression reveals the corresponding length.

7. 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.