Question

Use long division to find the quotient.
(6x^4-11x^3-7x^2+8x+3) / (3x+2)







A
2x3−5x2+x+12x^3-5x^2+x+12x
3
−5x
2
+x+1

B
2x3−5x2+x+2−13x+22x^3-5x^2+x+2-\frac{1}{3x+2}2x
3
−5x
2
+x+2−
3x+2
1
​


C
2x3−5x2+x+22x^3-5x^2+x+22x
3
−5x
2
+x+2

D
2x3−5x2+x+2+13x+22x^3-5x^2+x+2+\frac{1}{3x+2}2x
3
−5x
2
+x+2+
3x+2
1
​


(b)
Is (3x+2)
a factor of (6x^4-11x^3-7x^2+8x+3)


?

A
Yes

B
No

Answer 1

To perform the long division of \( (6x^4 - 11x^3 - 7x^2 + 8x + 3) \) by \( (3x + 2) \), we'll follow these steps:

1. **Divide the leading term:** Divide the leading term of the dividend \( 6x^4 \) by the leading term of the divisor \( 3x \):
\[
\frac{6x^4}{3x} = 2x^3
\]

2. **Multiply:** Multiply \( 2x^3 \) by the entire divisor \( (3x + 2) \):
\[
2x^3(3x + 2) = 6x^4 + 4x^3
\]

3. **Subtract:** Subtract this result from the original polynomial:
\[
(6x^4 - 11x^3 - 7x^2 + 8x + 3) - (6x^4 + 4x^3) = -15x^3 - 7x^2 + 8x + 3
\]

4. **Repeat the process:** Now repeat the process:
- Divide \( -15x^3 \) by \( 3x \):
\[
\frac{-15x^3}{3x} = -5x^2
\]
- Multiply \( -5x^2 \) by \( (3x + 2) \):
\[
-5x^2(3x + 2) = -15x^3 - 10x^2
\]
- Subtract this product from the current polynomial:
\[
(-15x^3 - 7x^2 + 8x + 3) - (-15x^3 - 10x^2) = 3x^2 + 8x + 3
\]

5. **Continue:** Repeat this one more time:
- Divide \( 3x^2 \) by \( 3x \):
\[
\frac{3x^2}{3x} = x
\]
- Multiply \( x \) by \( (3x + 2) \):
\[
x(3x + 2) = 3x^2 + 2x
\]
- Subtract:
\[
(3x^2 + 8x + 3) - (3x^2 + 2x) = 6x + 3
\]

6. **Final step:** Divide \( 6x \) by \( 3x \):
\[
\frac{6x}{3x} = 2
\]
- Multiply \( 2 \) by \( (3x + 2) \):
\[
2(3x + 2) = 6x + 4
\]
- Subtract:
\[
(6x + 3) - (6x + 4) = -1
\]

Thus, the final result is:
\[
2x^3 - 5x^2 + x + 2 - \frac{1}{3x + 2}
\]

So the quotient is \( 2x^3 - 5x^2 + x + 2 \) and the remainder is \( -1 \).

### Answer Choice
With this calculation, the answer that matches our quotient is:
**B** \( 2x^3 - 5x^2 + x + 2 - \frac{1}{3x + 2} \).

### Is \( (3x + 2) \) a factor of \( (6x^4 - 11x^3 - 7x^2 + 8x + 3) \)?
To determine if \( (3x + 2) \) is a factor, we can check if the remainder is zero.

Since we found the remainder to be \( -1 \), which is not zero, \( (3x + 2) \) is not a factor of \( (6x^4 - 11x^3 - 7x^2 + 8x + 3) \).

### Answer Choice
The answer is:
**B** No.