To divide the polynomial \( \frac{x^3 + 5x^2 - 7x + 2}{x - 2} \), we use polynomial long division. Here is the detailed, step-by-step process:
1. Set Up the Division:
Write \( x^3 + 5x^2 - 7x + 2 \) under the long division symbol and \( x - 2 \) outside of it.
2. Divide the First Term:
Divide the leading term of the numerator \( x^3 \) by the leading term of the denominator \( x \), which gives \( x^2 \).
3. Multiply and Subtract:
- Multiply \( x^2 \) by \( x - 2 \), which gives \( x^3 - 2x^2 \).
- Write this product under \( x^3 + 5x^2 \).
- Subtract \( x^3 - 2x^2 \) from \( x^3 + 5x^2 \), which results in \( 7x^2 - 7x + 2 \).
4. Repeat the Process:
- Divide the new leading term \( 7x^2 \) by \( x \), which gives \( 7x \).
- Multiply \( 7x \) by \( x - 2 \), which gives \( 7x^2 - 14x \).
- Subtract \( 7x^2 - 14x \) from \( 7x^2 - 7x + 2 \), which results in \( 7x + 2 \).
5. Continue the Division:
- Now divide \( 7x \) by \( x \), which gives \( 7 \).
- Multiply \( 7 \) by \( x - 2 \), which gives \( 7x - 14 \).
- Subtract \( 7x - 14 \) from \( 7x + 2 \), which results in \( 16 \).
6. Write the Result:
At this point, there are no more terms to bring down. The quotient is the result of all the divisions, and the remainder is the final result after the last subtraction. Therefore, the quotient is \( x^2 + 7x + 7 \) and the remainder is \( 16 \).
Thus, the result of the division is:
[tex]\[ \frac{x^3 + 5x^2 - 7x + 2}{x - 2} = x^2 + 7x + 7 + \frac{16}{x - 2} \][/tex]
So, the quotient is [tex]\( x^2 + 7x + 7 \)[/tex] and the remainder is [tex]\( 16 \)[/tex].
