I apologize for any confusion in my previous explanation. Let me clarify how to evaluate $P(x)$ at $x = -2$ for the polynomial $P(x) = -2x^4 - x^3 + 3x + 30$.

Given Polynomial

The polynomial given is: $P(x) = -2x^4 - x^3 + 3x + 30$

Evaluate $P(-2)$

To determine if $x + 2$ is a factor, we evaluate $P(x)$ at $x = -2$.

$P(-2) = -2(-2)^4 - (-2)^3 + 3(-2) + 30$

Step-by-Step Calculation

1. Calculate $(-2)^4$: $(-2)^4 = 16$

2. Calculate $-2(-2)^4$: $-2 \times 16 = -32$

3. Calculate $(-2)^3$: $(-2)^3 = -8$

4. Calculate $-(-2)^3$: $-(-8) = 8$

5. Calculate $3(-2)$: $3 \times (-2) = -6$

6. Combine all terms: $P(-2) = -32 + 8 - 6 + 30$

7. Simplify the expression: $P(-2) = (-32 + 8) + (-6 + 30) = -24 + 24 = 0$

Conclusion

Therefore, $P(-2) = 0$. This confirms that $x + 2$ is indeed a factor of the polynomial $P(x) = -2x^4 - x^3 + 3x + 30$.

Feel free to ask if you have any further questions or need additional clarification!

Related Questions

1. What is the Factor Theorem, and how is it used to determine factors of polynomials?
2. How can we find all factors of a given polynomial?
3. How can synthetic division be used to confirm the factorization of polynomials?
4. What is the remainder theorem, and how is it related to the factor theorem?
5. Can a polynomial have more than one linear factor?
6. How can the factor theorem help in finding roots of polynomial equations?
7. How can polynomial division be used to find the quotient when dividing by a factor?
8. What is the difference between the factor theorem and the division algorithm?

Tip

When evaluating polynomials, pay careful attention to the signs and powers in each term to ensure accurate calculations.