Given $$ P(x)=x^{3}+5 x^{2}+9 x+45 $$. Write $$ P $$ in factored form (as a product of linear factors). Be sure to write the full equation, including $$ P(x)= $$.

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UpStudy AI · Accepted Answer

We are given
$$
P(x)=x^3+5x^2+9x+45.
$$

**Step 1. Find a Rational Root**

To factor $$P(x)$$ as a product of linear factors, we start by finding a rational root. By the Rational Root Theorem, any rational root is a divisor of the constant term (45). Possible candidates include $$ \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45 $$.

We test $$ x=-5 $$:
$$
P(-5)=(-5)^3+5(-5)^2+9(-5)+45=-125+125-45+45=0.
$$
Thus, $$ x=-5 $$ is a root and $$ x+5 $$ is a factor.

**Step 2. Divide by $$ (x+5) $$**

Divide $$P(x)$$ by $$ x+5 $$ to find the quadratic factor.

Using synthetic division with root $$ -5 $$:

$$
\begin{array}{r|rrrr}
-5 & 1 & 5 & 9 & 45 \
   &   & -5 & 0 & -45 \
\hline
   & 1 & 0 & 9 & 0\
\end{array}
$$

The quotient is
$$
x^2+0x+9 = x^2+9.
$$

Thus, we have
$$
P(x) = (x+5)(x^2+9).
$$

**Step 3. Factor the Quadratic Further**

The quadratic $$x^2+9$$ can be factored over the complex numbers. Recall that
$$
x^2+9 = 0 \quad \Rightarrow \quad x^2 = -9 \quad \Rightarrow \quad x = \pm 3i.
$$
Thus,
$$
x^2+9=(x+3i)(x-3i).
$$

**Step 4. Write the Fully Factored Form**

Therefore, the complete factorization of $$P(x)$$ into linear factors (over the complex numbers) is:
$$
P(x) = (x+5)(x+3i)(x-3i).
$$
$$ P(x) = (x + 5)(x + 3i)(x - 3i) $$.

Given \( P(x)=x^{3}+5 x^{2}+9 x+45 \). Write \( P \) in factored form (as a product of linear factors). Be sure to write the full equation, including \( P(x)= \).

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