True or false
(x+2) is a factor of (x^3+8x^2-2x+3)

To determine if \((x+2)\) is a factor of \(x^3 + 8x^2 - 2x + 3\), we can apply synthetic division or use the factor theorem.

### Using the Factor Theorem:
The factor theorem states that \((x + a)\) is a factor of a polynomial \(P(x)\) if and only if \(P(-a) = 0\).

For \((x + 2)\) to be a factor, we substitute \(x = -2\) into the polynomial \(P(x) = x^3 + 8x^2 - 2x + 3\) and check if the result is 0.

\[
P(-2) = (-2)^3 + 8(-2)^2 - 2(-2) + 3
\]
\[
P(-2) = -8 + 8(4) + 4 + 3
\]
\[
P(-2) = -8 + 32 + 4 + 3 = 31
\]

Since \(P(-2) \neq 0\), \((x+2)\) is **not** a factor of \(x^3 + 8x^2 - 2x + 3\).

Thus, the statement is **false**.

Would you like more details on this process?

Here are some related questions to explore:

1. What is the Remainder Theorem and how is it applied in polynomial division?
2. How can synthetic division be used to divide polynomials?
3. What are the conditions for a polynomial to be divisible by a linear factor?
4. How can you find all factors of a cubic polynomial?
5. Can a polynomial have irrational or complex factors?

**Tip:** When verifying whether a polynomial has a specific factor, always check the result of substituting the root associated with the factor into the polynomial.

True or false (x+2) is a factor of (x^3+8x^2-2x+3)

Determine if (x+2) is a factor of the polynomial x^3 + 8x^2 - 2x + 3 using the factor theorem. This problem explores key algebraic concepts and synthetic division, ideal for students in Grades 9-12.

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