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Solve for \( x \).

[tex]\[ 3x = 6x - 2 \][/tex]



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6)
$
\begin{array}{l}
8 x^3+4 x^2-18 x-9 \\
P\left(-\frac{3}{2}\right)=8\left(-\frac{3}{2}\right)^3+
\end{array}
$
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Response:
6) Simplify the following expression:

[tex]\[
8x^3 + 4x^2 - 18x - 9
\][/tex]

Evaluate \( P \left( -\frac{3}{2} \right) \):

[tex]\[
P \left( -\frac{3}{2} \right) = 8 \left( -\frac{3}{2} \right)^3 + \ldots
\][/tex]



Answer :

GinnyAnswer
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].

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