To factor the polynomial [tex]\( x^3 + 4x^2 - 9x - 36 \)[/tex] completely, we want to express it as a product of simpler polynomials. Let's go through the steps:
1. Identify Rational Roots:
Look for possible rational roots of the polynomial by considering factors of the constant term (-36) and the leading coefficient (1). This involves checking possible values such as [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36 \)[/tex].
2. Test Rational Roots:
Substitute these values into the polynomial to see if they yield zero. The roots you find will help break down the polynomial into factors. Through this check, you will find that [tex]\( x = 3 \)[/tex], [tex]\( x = -3 \)[/tex], and [tex]\( x = -4 \)[/tex] are roots.
3. Form Factors:
Each root [tex]\( r \)[/tex] corresponds to a factor of [tex]\( (x - r) \)[/tex]. Therefore, the roots [tex]\( x = 3 \)[/tex], [tex]\( x = -3 \)[/tex], and [tex]\( x = -4 \)[/tex] give us the factors [tex]\( (x - 3) \)[/tex], [tex]\( (x + 3) \)[/tex], and [tex]\( (x + 4) \)[/tex].
4. Construct the Factored Form:
With the roots identified, we can write the polynomial as the product of its factors:
[tex]\[
x^3 + 4x^2 - 9x - 36 = (x - 3)(x + 3)(x + 4)
\][/tex]
Therefore, the completely factored form of the polynomial [tex]\( x^3 + 4x^2 - 9x - 36 \)[/tex] is:
[tex]\[
(x - 3)(x + 3)(x + 4)
\][/tex]
So among the provided options, the correct answer is:
[tex]\[
(x+3)(x-3)(x+4)
\][/tex]
