Math Problem Statement

Divide the polynomial (-a^3 + 25a) by (a - 5). If the division is not exact, include the remainder as a fraction.

Solution

The problem presented in the image asks for the division of the polynomial a3+25a-a^3 + 25a by (a5)(a - 5) and to include the remainder as a fraction if it doesn't divide evenly. Let's solve this using polynomial long division.

Step 1: Set up the division

We are dividing: (a3+0a2+25a+0)÷(a5)(-a^3 + 0a^2 + 25a + 0) \div (a - 5)

Step 2: Perform the division

  1. Divide the first term of the dividend a3-a^3 by the first term of the divisor aa: a3a=a2\frac{-a^3}{a} = -a^2

  2. Multiply a2-a^2 by (a5)(a - 5): a2(a5)=a3+5a2-a^2(a - 5) = -a^3 + 5a^2

  3. Subtract (a3+5a2)(-a^3 + 5a^2) from (a3+0a2)(-a^3 + 0a^2): (a3+0a2)(a3+5a2)=5a2(-a^3 + 0a^2) - (-a^3 + 5a^2) = -5a^2

  4. Divide the new first term 5a2-5a^2 by aa: 5a2a=5a\frac{-5a^2}{a} = -5a

  5. Multiply 5a-5a by (a5)(a - 5): 5a(a5)=5a2+25a-5a(a - 5) = -5a^2 + 25a

  6. Subtract (5a2+25a)(-5a^2 + 25a) from (5a2+25a)(-5a^2 + 25a): (5a2+25a)(5a2+25a)=0(-5a^2 + 25a) - (-5a^2 + 25a) = 0

Thus, the remainder is zero, and the quotient is: a25a-a^2 - 5a

Final Answer:

a25a\boxed{-a^2 - 5a}

Would you like a breakdown of any of the steps or further details?

Here are some related questions for you to consider:

  1. How is synthetic division different from polynomial long division?
  2. What happens if the divisor has a higher degree than the dividend?
  3. Can you apply the remainder theorem to check the remainder?
  4. What is the impact of changing the divisor in this division problem?
  5. How would this problem change if the coefficients of a2a^2 and aa were different?

Tip: Always ensure the dividend is written in standard form with all missing terms (e.g., 0a20a^2) before starting long division.

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Math Problem Analysis

Mathematical Concepts

Polynomial Division
Algebraic Long Division

Formulas

Polynomial Long Division

Theorems

Division Algorithm for Polynomials

Suitable Grade Level

Grades 9-12