Divide​ f(z) by​ d(z), and write a summary statement in polynomial form and fraction form.
​f(z)equals8z Superscript 4minus13z cubedplus58z squaredminus33zminus24​; ​d(z)equalsz squaredminuszplus7


We are tasked with dividing the polynomial \( f(z) = 8z^4 - 13z^3 + 58z^2 - 33z - 24 \) by \( d(z) = z^2 - z + 7 \), and expressing the result in both polynomial and fraction form.

To perform the division, we will use polynomial long division. Let's begin by dividing the leading terms.

1. **First step:** 
   Divide \( 8z^4 \) by \( z^2 \), which gives \( 8z^2 \).
   Multiply \( 8z^2 \) by \( d(z) = z^2 - z + 7 \), resulting in:
   \[
   8z^4 - 8z^3 + 56z^2
   \]
   Subtract this from \( f(z) \):
   \[
   (8z^4 - 13z^3 + 58z^2 - 33z - 24) - (8z^4 - 8z^3 + 56z^2) = -5z^3 + 2z^2 - 33z - 24
   \]

2. **Second step:**
   Divide \( -5z^3 \) by \( z^2 \), which gives \( -5z \).
   Multiply \( -5z \) by \( d(z) = z^2 - z + 7 \), resulting in:
   \[
   -5z^3 + 5z^2 - 35z
   \]
   Subtract this from the previous result:
   \[
   (-5z^3 + 2z^2 - 33z - 24) - (-5z^3 + 5z^2 - 35z) = -3z^2 + 2z - 24
   \]

3. **Third step:**
   Divide \( -3z^2 \) by \( z^2 \), which gives \( -3 \).
   Multiply \( -3 \) by \( d(z) = z^2 - z + 7 \), resulting in:
   \[
   -3z^2 + 3z - 21
   \]
   Subtract this from the previous result:
   \[
   (-3z^2 + 2z - 24) - (-3z^2 + 3z - 21) = -z - 3
   \]

Thus, the quotient is \( 8z^2 - 5z - 3 \) and the remainder is \( -z - 3 \).

### Summary in Polynomial and Fraction Form:
- **Polynomial form:** 
  \[
  f(z) = (z^2 - z + 7)(8z^2 - 5z - 3) + (-z - 3)
  \]
  
- **Fraction form:** 
  \[
  \frac{f(z)}{d(z)} = 8z^2 - 5z - 3 + \frac{-z - 3}{z^2 - z + 7}
  \]

Would you like any specific details or clarifications?

Here are some related questions:
1. How does polynomial long division work step-by-step?
2. What are some common applications of polynomial division?
3. How is synthetic division different from polynomial long division?
4. Can the remainder of polynomial division ever be zero?
5. How do we find the quotient and remainder when dividing polynomials?

**Tip:** Always align terms by powers of \( z \) when performing polynomial long division for accuracy!

Divide f(z) by d(z), and write a summary statement in polynomial form and fraction form. f(z) = 8z^4 - 13z^3 + 58z^2 - 33z - 24; d(z) = z^2 - z + 7

Explore the polynomial division of f(z) = 8z^4 - 13z^3 + 58z^2 - 33z - 24 by d(z) = z^2 - z + 7. This detailed step-by-step solution uses polynomial long division, resulting in both a quotient and remainder. Ideal for Grades 10-12 students studying algebra and polynomial theorems.

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