Question

Use long division to find the quotient.
x^3 + 4x^2 + 7x - 5 / x + 3

A. X^2 + 7x + 10 - 7 / x + 3
B. X^2 + X + 4 - 17 / x + 3
C. X^2 + x + 10 - 17 / x + 3
D. X^2 + 7x + 4 - 7 / x + 3

Answer 1

To divide the polynomial \( x^3 + 4x^2 + 7x - 5 \) by \( x + 3 \) using long division, we 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 the process**:
- Divide \( x^2 \) by \( x \) to get \( x \).
- Multiply \( x \) by \( x + 3 \) to get \( x^2 + 3x \).
- Subtract \( (x^2 + 3x) \):
\[
(x^2 + 7x - 5) - (x^2 + 3x) = (7x - 3x) - 5 = 4x - 5
\]

5. **Repeat again**:
- Divide \( 4x \) by \( x \) to get \( 4 \).
- Multiply \( 4 \) by \( x + 3 \) to get \( 4x + 12 \).
- Subtract \( (4x + 12) \):
\[
(4x - 5) - (4x + 12) = -5 - 12 = -17
\]

Now, we have completed the long division. The quotient is \( x^2 + x + 4 \), and the remainder is \( -17 \).

Thus, we can express the result of the division as:
\[
\frac{x^3 + 4x^2 + 7x - 5}{x + 3} = x^2 + x + 4 - \frac{17}{x + 3}
\]

Now, matching this with the options provided:

A. \( x^2 + 7x + 10 - \frac{7}{x + 3} \)

B. \( x^2 + x + 4 - \frac{17}{x + 3} \) **(this is correct)**

C. \( x^2 + x + 10 - \frac{17}{x + 3} \)

D. \( x^2 + 7x + 4 - \frac{7}{x + 3} \)

The correct answer is **B**.