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
Answers
GPT-4o mini
To perform long division of the polynomial \( x^3 + 4x^2 + 7x - 5 \) by \( x + 3 \), follow these steps:
1. **Divide the first term**: Divide \( x^3 \) by \( x \) to get \( x^2 \).
2. **Multiply**: Multiply \( x^2 \) by \( x + 3 \) to get \( x^3 + 3x^2 \).
3. **Subtract**: Subtract \( (x^3 + 3x^2) \) from \( (x^3 + 4x^2 + 7x - 5) \):
\[
(x^3 + 4x^2 + 7x - 5) - (x^3 + 3x^2) = (4x^2 - 3x^2) + 7x - 5 = x^2 + 7x - 5.
\]
4. **Repeat**: Now divide \( x^2 \) by \( x \) to get \( x \).
5. **Multiply**: Multiply \( x \) by \( x + 3 \) to get \( x^2 + 3x \).
6. **Subtract**: Subtract \( (x^2 + 3x) \) from \( (x^2 + 7x - 5) \):
\[
(x^2 + 7x - 5) - (x^2 + 3x) = (7x - 3x) - 5 = 4x - 5.
\]
7. **Repeat**: Now divide \( 4x \) by \( x \) to get \( 4 \).
8. **Multiply**: Multiply \( 4 \) by \( x + 3 \) to get \( 4x + 12 \).
9. **Subtract**: Subtract \( (4x + 12) \) from \( (4x - 5) \):
\[
(4x - 5) - (4x + 12) = -5 - 12 = -17.
\]
At this point, we have exhausted the polynomial, so we write the quotient and remainder:
- The quotient is \( x^2 + x + 4 \).
- The remainder is \( -17 \).
So, the final answer can be expressed as:
\[
x^2 + x + 4 - \frac{17}{x + 3}.
\]
Thus, the correct response from your choices is:
**x² + x + 4 − 17/(x + 3)**.
1. **Divide the first term**: Divide \( x^3 \) by \( x \) to get \( x^2 \).
2. **Multiply**: Multiply \( x^2 \) by \( x + 3 \) to get \( x^3 + 3x^2 \).
3. **Subtract**: Subtract \( (x^3 + 3x^2) \) from \( (x^3 + 4x^2 + 7x - 5) \):
\[
(x^3 + 4x^2 + 7x - 5) - (x^3 + 3x^2) = (4x^2 - 3x^2) + 7x - 5 = x^2 + 7x - 5.
\]
4. **Repeat**: Now divide \( x^2 \) by \( x \) to get \( x \).
5. **Multiply**: Multiply \( x \) by \( x + 3 \) to get \( x^2 + 3x \).
6. **Subtract**: Subtract \( (x^2 + 3x) \) from \( (x^2 + 7x - 5) \):
\[
(x^2 + 7x - 5) - (x^2 + 3x) = (7x - 3x) - 5 = 4x - 5.
\]
7. **Repeat**: Now divide \( 4x \) by \( x \) to get \( 4 \).
8. **Multiply**: Multiply \( 4 \) by \( x + 3 \) to get \( 4x + 12 \).
9. **Subtract**: Subtract \( (4x + 12) \) from \( (4x - 5) \):
\[
(4x - 5) - (4x + 12) = -5 - 12 = -17.
\]
At this point, we have exhausted the polynomial, so we write the quotient and remainder:
- The quotient is \( x^2 + x + 4 \).
- The remainder is \( -17 \).
So, the final answer can be expressed as:
\[
x^2 + x + 4 - \frac{17}{x + 3}.
\]
Thus, the correct response from your choices is:
**x² + x + 4 − 17/(x + 3)**.