Let's solve the polynomial \( P(x) = 8x^3 + 4x^2 - 18x - 9 \) for \( x = -\frac{3}{2} \).
First, we break it down into parts and evaluate each term separately:
### Step 1: Evaluate \( 8x^3 \)
We need to calculate \( 8 \left( -\frac{3}{2} \right)^3 \):
[tex]\[
\left( -\frac{3}{2} \right)^3 = \left( -\frac{3}{2} \right) \times \left( -\frac{3}{2} \right) \times \left( -\frac{3}{2} \right) = -\frac{27}{8}
\][/tex]
So,
[tex]\[
8 \left( -\frac{3}{2} \right)^3 = 8 \times -\frac{27}{8} = -27
\][/tex]
Thus, the first term is \(-27\).
### Step 2: Evaluate \( 4x^2 \)
Next, calculate \( 4 \left( -\frac{3}{2} \right)^2 \):
[tex]\[
\left( -\frac{3}{2} \right)^2 = \left( -\frac{3}{2} \right) \times \left( -\frac{3}{2} \right) = \frac{9}{4}
\][/tex]
So,
[tex]\[
4 \left( -\frac{3}{2} \right)^2 = 4 \times \frac{9}{4} = 9
\][/tex]
Thus, the second term is \(9\).
### Step 3: Evaluate \( -18x \)
Next, calculate \( -18 \left( -\frac{3}{2} \right) \):
[tex]\[
-18 \left( -\frac{3}{2} \right) = 27
\][/tex]
Thus, the third term is \(27\).
### Step 4: Evaluate the constant term
The constant term does not change, it remains \(-9\).
### Step 5: Sum all terms
Finally, sum all the terms to get \( P\left( -\frac{3}{2} \right) \):
[tex]\[
-27 + 9 + 27 - 9 = 0
\][/tex]
### Conclusion
So, the value of the polynomial \( P\left( -\frac{3}{2} \right) \) is \(0\).
To summarize:
- The first term \( 8 \left( -\frac{3}{2} \right)^3 = -27 \)
- The second term \( 4 \left( -\frac{3}{2} \right)^2 = 9 \)
- The third term \( -18 \left( -\frac{3}{2} \right) = 27 \)
- The sum is \(-27 + 9 + 27 - 9 = 0\)
So, the final value of [tex]\( P\left( -\frac{3}{2} \right) \)[/tex] is [tex]\(0\)[/tex].
