To find [tex]\( P(-2) \)[/tex] for [tex]\( P(x) = -2x^3 - 4x^2 - 9 \)[/tex] using the remainder theorem, we follow these steps:
1. Identify the polynomial:
[tex]\[
P(x) = -2x^3 - 4x^2 - 9
\][/tex]
2. Use Polynomial Long Division (or Synthetic Division):
We need to divide [tex]\( P(x) \)[/tex] by [tex]\( (x + 2) \)[/tex] to find the quotient and the remainder.
We start the division:
[tex]\[
\frac{-2x^3 - 4x^2 - 9}{x + 2}
\][/tex]
3. Perform the division step-by-step:
- The leading term of the quotient is determined by dividing the leading term of the dividend by the leading term of the divisor:
[tex]\[
-2x^3 \div x = -2x^2
\][/tex]
- Multiply the entire divisor by this leading term:
[tex]\[
(-2x^2)(x + 2) = -2x^3 - 4x^2
\][/tex]
- Subtract this from the original polynomial:
[tex]\[
(-2x^3 - 4x^2 - 9) - (-2x^3 - 4x^2) = -9
\][/tex]
At this point, the remainder is [tex]\(-9\)[/tex], and we have no more terms to divide.
So, the quotient is:
[tex]\[
-2x^2
\][/tex]
And the remainder is:
[tex]\[
-9
\][/tex]
4. Summary of Division:
The quotient of the division is:
[tex]\[
\text{Quotient} = -2x^2
\][/tex]
The remainder of the division is:
[tex]\[
\text{Remainder} = -9
\][/tex]
5. Apply the Remainder Theorem:
According to the Remainder Theorem, the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( x + 2 \)[/tex] is [tex]\( P(-2) \)[/tex].
Therefore,
[tex]\[
P(-2) = -9
\][/tex]
Thus, the final answer is:
[tex]\[
\text{Quotient} = -2x^2
\][/tex]
[tex]\[
\text{Remainder} = -9
\][/tex]
[tex]\[
P(-2) = -9
\][/tex]
