Question

given $p(x) = x^3 - 3x^2 - 2x + 4$, which statement is true?

1. $(x - 1)$ is a factor because $p(-1) = 2$.
2. $(x + 1)$ is a factor because $p(-1) = 2$.
3. $(x + 1)$ is a factor because $p(1) = 0$.
4. $(x - 1)$ is a factor because $p(1) = 0$.

Explanation:

First, correct the polynomial: assuming the last term is 4 (common in such problems), so \(P(x) = x^3 - 3x^2 - 2x + 4\).

Check each option:

• Factor Theorem: \((x - a)\) is a factor iff \(P(a) = 0\).
• Option 1: \((x-1)\) factor? Check \(P(1)\):

\(P(1) = 1^3 -3(1)^2 -2(1) +4 =1 -3 -2 +4=0\). But option 1 says \(P(-1)=2\) (wrong reason), so invalid.

• Option 2: \((x+1)\) factor? Check \(P(-1)\):

\(P(-1)=(-1)^3 -3(-1)^2 -2(-1)+4=-1 -3 +2 +4=2≠0\), so not a factor. Invalid.

• Option3: \((x+1)\) factor? \(P(1)=0\) (irrelevant to \(x+1\)), invalid.
• Option4: \((x-1)\) factor because \(P(1)=0\): Correct (matches Factor Theorem).

Step1: Apply Factor Theorem

\((x-a)\) is factor iff \(P(a)=0\)

Step2: Calculate \(P(1)\)

\(P(1)=1-3-2+4=0\)

Step3: Verify option 4

\((x-1)\) is factor since \(P(1)=0\) (matches option4)

Answer:

1. (x - 1) is a factor because P(1) = 0