To factor the polynomial [tex]\( f(x) = x^5 - 5x^4 - 13x^3 + 65x^2 + 36x - 180 \)[/tex] completely, we are given that [tex]\( x = 5 \)[/tex] is a root. Knowing that [tex]\( x = 5 \)[/tex] is a root, we can factor out [tex]\( (x - 5) \)[/tex] from [tex]\( f(x) \)[/tex]. Let's proceed with the step-by-step factorization:
### Step 1: Polynomial Division
Since [tex]\( x = 5 \)[/tex] is a root of [tex]\( f(x) \)[/tex], [tex]\( x - 5 \)[/tex] is a factor of [tex]\( f(x) \)[/tex]. We perform polynomial division of [tex]\( f(x) \)[/tex] by [tex]\( x - 5 \)[/tex].
1. Setup the problem:
[tex]\( f(x) = x^5 - 5x^4 - 13x^3 + 65x^2 + 36x - 180 \)[/tex]
We divide this by [tex]\( x - 5 \)[/tex].
2. Perform polynomial division:
By synthetic division or long division, we divide the polynomial by [tex]\( x - 5 \)[/tex]:
[tex]\[
\begin{array}{r|rrrrrr}
5 & 1 & -5 & -13 & 65 & 36 & -180 \\
\hline
& 1 & 0 & -13 & 0 & 36 & 0 \\
\end{array}
\][/tex]
We get the quotient as:
[tex]\[
x^4 - 13x^2 + 36
\][/tex]
So, we have:
[tex]\[
f(x) = (x - 5)(x^4 - 13x^2 + 36)
\][/tex]
### Step 2: Factor the Quotient Polynomial
Next, we need to factor the quotient polynomial [tex]\( x^4 - 13x^2 + 36 \)[/tex].
We recognize it as a quadratic in terms of [tex]\( x^2 \)[/tex]. Let's rewrite [tex]\( x^4 - 13x^2 + 36 \)[/tex] as:
[tex]\[
y = x^2
\][/tex]
[tex]\[
y^2 - 13y + 36
\][/tex]
### Step 3: Solve the Quadratic [tex]\( y^2 - 13y + 36 \)[/tex]
To factor [tex]\( y^2 - 13y + 36 \)[/tex], we find the roots of the quadratic equation:
[tex]\[
y^2 - 13y + 36 = 0
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{13 \pm \sqrt{13^2 - 4 \cdot 36}}{2}
\][/tex]
[tex]\[
y = \frac{13 \pm \sqrt{169 - 144}}{2}
\][/tex]
[tex]\[
y = \frac{13 \pm \sqrt{25}}{2}
\][/tex]
[tex]\[
y = \frac{13 \pm 5}{2}
\][/tex]
So, we have:
[tex]\[
y = 9 \quad \text{or} \quad y = 4
\][/tex]
In terms of [tex]\( x \)[/tex],
[tex]\[
x^2 = 9 \quad \text{or} \quad x^2 = 4
\][/tex]
[tex]\[
x = \pm 3 \quad \text{or} \quad x = \pm 2
\][/tex]
### Step 4: Write the Final Factorization
Therefore, we can factor [tex]\( x^4 - 13x^2 + 36 \)[/tex] as:
[tex]\[
x^4 - 13x^2 + 36 = (x^2 - 9)(x^2 - 4) = (x - 3)(x + 3)(x - 2)(x + 2)
\][/tex]
### Step 5: Combine All Factors
Taking all the factors into account, the completely factored form of [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = (x - 5)(x - 3)(x - 2)(x + 2)(x + 3)
\][/tex]
So, the polynomial [tex]\( f(x) \)[/tex] is factored completely as:
[tex]\[
f(x) = (x - 5)(x - 3)(x - 2)(x + 2)(x + 3)
\][/tex]
