What are the steps to solving this? (x
5
– 4x
3
+ 4x
2
) ÷ (x – 4)

To solve \(\frac{x^5 - 4x^3 + 4x^2}{x - 4}\), we'll use polynomial division. Here’s a step-by-step guide:

1. **Set Up the Division**:
   Arrange the dividend (the polynomial in the numerator) in descending powers of \(x\). If any term is missing, insert it with a coefficient of zero.
   
   So, we rewrite \(x^5 - 4x^3 + 4x^2\) as:
   \[
   x^5 + 0x^4 - 4x^3 + 4x^2 + 0x + 0
   \]

2. **Divide the Leading Terms**:
   Divide the leading term of the dividend (\(x^5\)) by the leading term of the divisor (\(x\)):
   \[
   \frac{x^5}{x} = x^4
   \]
   Write \(x^4\) as the first term of the quotient.

3. **Multiply and Subtract**:
   Multiply \(x^4\) by the divisor \((x - 4)\) and subtract the result from the dividend:
   \[
   (x^4)(x - 4) = x^5 - 4x^4
   \]
   Subtract this from the original polynomial:
   \[
   (x^5 + 0x^4 - 4x^3 + 4x^2) - (x^5 - 4x^4) = 4x^4 - 4x^3 + 4x^2
   \]

4. **Repeat the Process**:
   - **Next term**: Divide \(4x^4\) by \(x\), giving \(4x^3\).
   - Multiply \(4x^3\) by \(x - 4\):
     \[
     (4x^3)(x - 4) = 4x^4 - 16x^3
     \]
   - Subtract:
     \[
     (4x^4 - 4x^3 + 4x^2) - (4x^4 - 16x^3) = 12x^3 + 4x^2
     \]
   - Continue this process until the remainder is zero or the degree of the remainder is less than the degree of the divisor.

5. **Write the Quotient and Remainder**:
   Continue the division steps until no further terms can be divided by \(x\), resulting in the quotient polynomial and any remainder.

Would you like me to continue with the calculations to completion?

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### Related Practice Questions

1. How do you perform polynomial division with missing terms?
2. What happens if the divisor is a higher degree than the dividend?
3. Can synthetic division be used for polynomials with non-monomial divisors?
4. What role does the remainder play in polynomial division?
5. How can you check your answer after completing polynomial division?

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**Tip**: Always check each subtraction carefully in polynomial division, as it’s easy to make sign errors that affect the rest of the solution.

What are the steps to solving this? (x^5 - 4x^3 + 4x^2) ÷ (x - 4)

Discover the step-by-step solution for dividing the polynomial (x^5 - 4x^3 + 4x^2) by (x - 4) using polynomial division. This guide covers polynomial division concepts, essential formulas, and detailed calculations, suitable for students in Grades 10-12.

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