long divide (x^5-3x^4+10x^3-8x+75)/(x^2-5)

Let's perform the long division of \( \frac{x^5 - 3x^4 + 10x^3 - 8x + 75}{x^2 - 5} \) step by step.

### Step 1: Setup the division
We divide the polynomial \( x^5 - 3x^4 + 10x^3 - 8x + 75 \) by \( x^2 - 5 \).

### Step 2: First division
We divide the leading term of the dividend \( x^5 \) by the leading term of the divisor \( x^2 \):

\[
\frac{x^5}{x^2} = x^3
\]

Multiply \( x^3 \) by \( x^2 - 5 \):

\[
x^3 \cdot (x^2 - 5) = x^5 - 5x^3
\]

Subtract this from the original polynomial:

\[
(x^5 - 3x^4 + 10x^3 - 8x + 75) - (x^5 - 5x^3) = -3x^4 + 15x^3 - 8x + 75
\]

### Step 3: Second division
Now, divide \( -3x^4 \) by \( x^2 \):

\[
\frac{-3x^4}{x^2} = -3x^2
\]

Multiply \( -3x^2 \) by \( x^2 - 5 \):

\[
-3x^2 \cdot (x^2 - 5) = -3x^4 + 15x^2
\]

Subtract this from the new polynomial:

\[
(-3x^4 + 15x^3 - 8x + 75) - (-3x^4 + 15x^2) = 15x^3 - 15x^2 - 8x + 75
\]

### Step 4: Third division
Next, divide \( 15x^3 \) by \( x^2 \):

\[
\frac{15x^3}{x^2} = 15x
\]

Multiply \( 15x \) by \( x^2 - 5 \):

\[
15x \cdot (x^2 - 5) = 15x^3 - 75x
\]

Subtract this from the current polynomial:

\[
(15x^3 - 15x^2 - 8x + 75) - (15x^3 - 75x) = -15x^2 + 67x + 75
\]

### Step 5: Fourth division
Now, divide \( -15x^2 \) by \( x^2 \):

\[
\frac{-15x^2}{x^2} = -15
\]

Multiply \( -15 \) by \( x^2 - 5 \):

\[
-15 \cdot (x^2 - 5) = -15x^2 + 75
\]

Subtract this from the current polynomial:

\[
(-15x^2 + 67x + 75) - (-15x^2 + 75) = 67x
\]

### Step 6: Final result
At this point, we can't divide further, so \( 67x \) is the remainder.

The final answer is:

\[
\frac{x^5 - 3x^4 + 10x^3 - 8x + 75}{x^2 - 5} = x^3 - 3x^2 + 15x - 15 + \frac{67x}{x^2 - 5}
\]

Would you like further clarification on any part of this process?

Here are some related questions:
1. How do you perform synthetic division for polynomials?
2. Can long division be used for non-polynomial expressions?
3. What are common errors when performing polynomial long division?
4. How do you check the result of polynomial division?
5. What is the relationship between polynomial division and the remainder theorem?

**Tip:** Always align terms by their degree during polynomial division to avoid errors.

Perform a detailed step-by-step long division of the polynomial (x^5-3x^4+10x^3-8x+75) by (x^2-5). Learn key concepts in polynomial division, including dividing, multiplying, subtracting, and understanding the remainder. This problem is ideal for high school students in Grades 10-12.

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