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 cubed minus 5 x squared plus 4 x minus 8

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

x3+4x2−6x+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 \(x - 2\) without a remainder, we can use polynomial long division for each polynomial. Alternatively, we can also use the Remainder Theorem, which states that if a polynomial \(f(x)\) is evaluated at \(x = c\), the remainder of the division of \(f(x)\) by \(x - c\) is simply \(f(c)\). If \(f(c) = 0\), then the polynomial is divisible by \(x - c\).

We will evaluate each polynomial at \(x = 2\):

1. **Polynomial:** \(2x^3 - 5x^2 + 4x - 8\)

\[
f(2) = 2(2^3) - 5(2^2) + 4(2) - 8
\]
\[
= 2(8) - 5(4) + 8 - 8
\]
\[
= 16 - 20 + 8 - 8 = -4
\]

*Not divisible by \(x - 2\)*.

2. **Polynomial:** \(x^3 + 2x^2 - 5x - 6\)

\[
f(2) = (2^3) + 2(2^2) - 5(2) - 6
\]
\[
= 8 + 2(4) - 10 - 6
\]
\[
= 8 + 8 - 10 - 6 = 0
\]

*Divisible by \(x - 2\)*.

3. **Polynomial:** \(x^3 + 4x^2 - 6x + 8\)

\[
f(2) = (2^3) + 4(2^2) - 6(2) + 8
\]
\[
= 8 + 4(4) - 12 + 8
\]
\[
= 8 + 16 - 12 + 8 = 20
\]

*Not divisible by \(x - 2\)*.

4. **Polynomial:** \(3x^3 + 7x^2 - 2x + 5\)

\[
f(2) = 3(2^3) + 7(2^2) - 2(2) + 5
\]
\[
= 3(8) + 7(4) - 4 + 5
\]
\[
= 24 + 28 - 4 + 5 = 53
\]

*Not divisible by \(x - 2\)*.

After evaluating all polynomials, we find that:

**The polynomial \(x^3 + 2x^2 - 5x - 6\) is divisible by \(x - 2\) without a remainder.**