QuestionJuly 14, 2025

Part III: Use the Remainder Theorem to explain whether or not (x-2) is a factor of F(x)=x^4-2x^3+3x^2-10x+3 (3 points)

Part III: Use the Remainder Theorem to explain whether or not (x-2) is a factor of F(x)=x^4-2x^3+3x^2-10x+3 (3 points)
Part III: Use the Remainder Theorem to explain whether or not (x-2) is a factor of F(x)=x^4-2x^3+3x^2-10x+3 (3 points)

Solution
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Answer

(x-2) is not a factor of F(x). Explanation 1. Apply the Remainder Theorem The Remainder Theorem states that if x - c is a factor of F(x), then F(c) = 0. Here, c = 2. 2. Evaluate F(2) Substitute x = 2 into F(x): F(2) = 2^4 - 2 \cdot 2^3 + 3 \cdot 2^2 - 10 \cdot 2 + 3. 3. Simplify the Expression Calculate each term: 16 - 16 + 12 - 20 + 3 = -5. 4. Determine Factor Status Since F(2) \neq 0, (x-2) is not a factor of F(x).

Explanation

1. Apply the Remainder Theorem<br /> The Remainder Theorem states that if $x - c$ is a factor of $F(x)$, then $F(c) = 0$. Here, $c = 2$.<br /><br />2. Evaluate $F(2)$<br /> Substitute $x = 2$ into $F(x)$: <br /> $F(2) = 2^4 - 2 \cdot 2^3 + 3 \cdot 2^2 - 10 \cdot 2 + 3$.<br /><br />3. Simplify the Expression<br /> Calculate each term: $16 - 16 + 12 - 20 + 3 = -5$.<br /><br />4. Determine Factor Status<br /> Since $F(2) \neq 0$, $(x-2)$ is not a factor of $F(x)$.
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