Use the factor theorem to determine whether x+2 is a factor of $$ P\left(x\right)-2x^4-x^3+3x+30 $$specifically, evaluate P at the proper value, and then determine whether x+2 is a factor

To determine whether $ x + 2 $ is a factor of the polynomial $ P(x) = -2x^4 - x^3 + 3x + 30 $, we can use the **Factor Theorem**. The theorem states that $ x - c $ is a factor of a polynomial $ P(x) $ if and only if $ P(c) = 0 $.

In this case, we need to determine if $ x + 2 $ is a factor, which means we should evaluate the polynomial at $ x = -2 $ (since $ x + 2 = 0 $ when $ x = -2 $).

### Step-by-Step Solution

1. **Substitute $ x = -2 $ into $ P(x) $:**

   \[
   P(-2) = -2(-2)^4 - (-2)^3 + 3(-2) + 30
   \]

2. **Calculate each term:**

   - $(-2)^4 = 16$
   - $-2(-2)^4 = -2 \times 16 = -32$
   - $(-2)^3 = -8$
   - $-(-2)^3 = 8$
   - $3(-2) = -6$

3. **Evaluate $ P(-2) $:**

   \[
   P(-2) = -32 + 8 - 6 + 30
   \]

4. **Simplify the expression:**

   \[
   P(-2) = (-32 + 8) + (-6 + 30) = -24 + 24 = 0
   \]

Since $ P(-2) = 0 $, by the Factor Theorem, $ x + 2 $ is indeed a factor of the polynomial $ P(x) = -2x^4 - x^3 + 3x + 30 $.

### Conclusion

The factor theorem confirms that $ x + 2 $ is a factor of the polynomial $ P(x) = -2x^4 - x^3 + 3x + 30 $ because evaluating the polynomial at $ x = -2 $ results in zero.

Would you like to see a more detailed explanation, or do you have any questions about this process?

### 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

Remember that when using the factor theorem, it is crucial to substitute the correct value of $ x $ that corresponds to the zero of the potential factor. In this case, for $ x + 2 $, you substitute $ x = -2 $.

Explore how to use the Factor Theorem to determine whether x + 2 is a factor of the polynomial P(x) = -2x^4 - x^3 + 3x + 30. Step-by-step solution and detailed explanation included.

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