Question

Use long division to determine which of the following polynomials is divisible by x - 2 without a remainder. (4 points)
Responses

2x3−5x2+4x−8
2
𝑥
3
−
5
𝑥
2
+
4
𝑥
−
8
2 x cubed minus 5 x squared plus 4 x minus 8

x3+2x2−5x−6
𝑥
3
+
2
𝑥
2
−
5
𝑥
−
6
x cubed plus 2 x squared minus 5 x minus 6

x3+4x2−6x+8
𝑥
3
+
4
𝑥
2
−
6
𝑥
+
8
x cubed plus 4 x squared minus 6 x plus 8

3x3+7x2−2x+5

Answer 1

To determine which of the given polynomials is divisible by \( x - 2 \) using long division, we will perform polynomial long division for each of the four expressions and see if the remainder is zero. A polynomial is divisible by \( x - 2 \) if plugging \( x = 2 \) into the polynomial yields zero (Remainder Theorem).

Alternatively, we can directly use the Remainder Theorem to evaluate each polynomial at \( x = 2 \) to check for divisibility.

1. **Polynomial: \( 2x^3 - 5x^2 + 4x - 8 \)**
- Calculate \( P(2) \):
\[
P(2) = 2(2^3) - 5(2^2) + 4(2) - 8
\]
\[
= 2(8) - 5(4) + 8 - 8
\]
\[
= 16 - 20 + 8 - 8 = -4
\]
Not divisible.

2. **Polynomial: \( x^3 + 2x^2 - 5x - 6 \)**
- Calculate \( P(2) \):
\[
P(2) = (2^3) + 2(2^2) - 5(2) - 6
\]
\[
= 8 + 2(4) - 10 - 6
\]
\[
= 8 + 8 - 10 - 6 = 0
\]
Divisible.

3. **Polynomial: \( x^3 + 4x^2 - 6x + 8 \)**
- Calculate \( P(2) \):
\[
P(2) = (2^3) + 4(2^2) - 6(2) + 8
\]
\[
= 8 + 4(4) - 12 + 8
\]
\[
= 8 + 16 - 12 + 8 = 20
\]
Not divisible.

4. **Polynomial: \( 3x^3 + 7x^2 - 2x + 5 \)**
- Calculate \( P(2) \):
\[
P(2) = 3(2^3) + 7(2^2) - 2(2) + 5
\]
\[
= 3(8) + 7(4) - 4 + 5
\]
\[
= 24 + 28 - 4 + 5 = 53
\]
Not divisible.

### Conclusion
The polynomial that is divisible by \( x - 2 \) without a remainder is:
\[
\boxed{x^3 + 2x^2 - 5x - 6}
\]