Factor the following polynomial equation: $$ f(x)=8 x^{3}-27 $$ $$ f(x)=(a x+b)\left(c x^{2}+d x+e\right) $$, where $$ a=\square $$ $$ b=\square $$ $$ c=\square $$ $$ d=\square $$

Question

UpStudy AI · Accepted Answer

To factor the polynomial $$ f(x) = 8x^{3} - 27 $$, we can recognize it as a **difference of cubes**.

### Step-by-Step Factoring:

1. **Identify the cubes:**
   $$
   8x^3 = (2x)^3 \quad \text{and} \quad 27 = 3^3
   $$

2. **Use the difference of cubes formula:**
   $$
   A^3 - B^3 = (A - B)(A^2 + AB + B^2)
   $$
   Here, $$ A = 2x $$ and $$ B = 3 $$.

3. **Apply the formula:**
   $$
   8x^3 - 27 = (2x - 3)\left((2x)^2 + (2x)(3) + 3^2\right)
   $$
   $$
   = (2x - 3)(4x^2 + 6x + 9)
   $$

### Matching to the Given Form:

The factored form is:
$$
f(x) = (2x - 3)(4x^2 + 6x + 9)
$$

So, comparing to $$ f(x) = (a x + b)(c x^2 + d x + e) $$:

$$
\begin{cases}
a = 2 \
b = -3 \
c = 4 \
d = 6 \
e = 9 \
\end{cases}
$$

### Final Answer:

$$
\boxed{
\begin{aligned}
a &= 2 \
b &= -3 \
c &= 4 \
d &= 6 \
e &= 9 \
\end{aligned}
}
$$
$$ a = 2 $$, $$ b = -3 $$, $$ c = 4 $$, $$ d = 6 $$, $$ e = 9 $$.

Factor the following polynomial equation: \( f(x)=8 x^{3}-27 \) \( f(x)=(a x+b)\left(c x^{2}+d x+e\right) \), where \( a=\square \) \( b=\square \) \( c=\square \) \( d=\square \)

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