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

Answers

Answered by GPT-4o mini
To determine which of the given polynomials is divisible by \( x - 2 \) without a remainder, we can perform polynomial long division on each polynomial with \( x - 2 \) as the divisor, or we can evaluate each polynomial at \( x = 2 \). If substituting \( x = 2 \) into the polynomial results in zero, then the polynomial is divisible by \( x - 2 \).

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

1. **For \( 2x^3 - 5x^2 + 4x - 8 \)**:
\[
2(2)^3 - 5(2)^2 + 4(2) - 8 = 2(8) - 5(4) + 8 - 8 = 16 - 20 + 8 - 8 = -4 \quad (\text{Not divisible})
\]

2. **For \( x^3 + 2x^2 - 5x - 6 \)**:
\[
(2)^3 + 2(2)^2 - 5(2) - 6 = 8 + 2(4) - 10 - 6 = 8 + 8 - 10 - 6 = 0 \quad (\text{Divisible})
\]

3. **For \( x^3 + 4x^2 - 6x + 8 \)**:
\[
(2)^3 + 4(2)^2 - 6(2) + 8 = 8 + 4(4) - 12 + 8 = 8 + 16 - 12 + 8 = 20 \quad (\text{Not divisible})
\]

4. **For \( 3x^3 + 7x^2 - 2x + 5 \)**:
\[
3(2)^3 + 7(2)^2 - 2(2) + 5 = 3(8) + 7(4) - 4 + 5 = 24 + 28 - 4 + 5 = 53 \quad (\text{Not divisible})
\]

Based on the evaluations, the only polynomial that is divisible by \( x - 2 \) without a remainder is:

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