Question

Use long division to find the quotient. x3+4x2+7x−5x+3 (1 point) Responses x2+7x+10−7x+3 x squared plus 7 x plus 10 minus the fraction with numerator 7 and denominator x plus 3 x2+x+4−17x+3 x squared plus x plus 4 minus the fraction with numerator 17 and denominator x plus 3 x2+x+10−17x+3 x squared plus x plus 10 minus the fraction with numerator 17 and denominator x plus 3 x2+7x+4−7x+3

Answer 1

To perform long division on the expression \( x^3 + 4x^2 + 7x - 5 \) by \( x + 3 \), we will follow the steps for polynomial long division.

1. **Divide the leading term**: Divide the leading term of the dividend, \( x^3 \), by the leading term of the divisor, \( x \):
\[
x^3 \div x = x^2
\]

2. **Multiply and subtract**: Multiply \( x^2 \) by the entire divisor \( x + 3 \) and subtract it from the dividend.
\[
(x^3 + 4x^2 + 7x - 5) - (x^2(x + 3)) = (x^3 + 4x^2 + 7x - 5) - (x^3 + 3x^2)
\]
This simplifies to:
\[
(4x^2 - 3x^2) + 7x - 5 = x^2 + 7x - 5
\]

3. **Repeat**: Now, repeat the process with \( x^2 + 7x - 5 \).
- Divide \( x^2 \) by \( x \):
\[
x^2 \div x = x
\]
- Multiply and subtract:
\[
(x^2 + 7x - 5) - (x(x + 3)) = (x^2 + 7x - 5) - (x^2 + 3x)
\]
This simplifies to:
\[
(7x - 3x) - 5 = 4x - 5
\]

4. **Continue to the next term**: Now, repeat with \( 4x - 5 \).
- Divide \( 4x \) by \( x \):
\[
4x \div x = 4
\]
- Multiply and subtract:
\[
(4x - 5) - (4(x + 3)) = (4x - 5) - (4x + 12)
\]
This simplifies to:
\[
-5 - 12 = -17
\]

5. **Final result**: The quotient is \( x^2 + x + 4 \) and the remainder is \( -17 \). So, we can write this as:
\[
\frac{x^3 + 4x^2 + 7x - 5}{x + 3} = x^2 + x + 4 - \frac{17}{x + 3}
\]

Hence, the correct response is:
**\( x^2 + x + 4 - \frac{17}{x + 3} \)**.