To factor the polynomial [tex]\( f(x) = x^3 - 5x^2 - 18x + 72 \)[/tex] as a product of linear factors, follow these steps:
1. Identify the Polynomial:
We start with the polynomial [tex]\( f(x) = x^3 - 5x^2 - 18x + 72 \)[/tex].
2. Find the Roots of the Polynomial:
To factor the polynomial, we need to find its roots. These roots are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. Let's denote these roots by [tex]\( r_1, r_2, \)[/tex] and [tex]\( r_3 \)[/tex].
3. Check for Possible Rational Roots:
We know that [tex]\( f(x) = 0 \)[/tex] at the roots [tex]\( r_1, r_2, \)[/tex] and [tex]\( r_3 \)[/tex]. By examining possible rational roots (factors of the constant term 72 and the leading coefficient 1), we can find that the roots are [tex]\( x = 6 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = -4 \)[/tex].
4. Express the Polynomial in Factored Form:
Once the roots are determined, we can write the polynomial as a product of linear factors corresponding to these roots. If [tex]\( r_1 = 6 \)[/tex], [tex]\( r_2 = 3 \)[/tex], and [tex]\( r_3 = -4 \)[/tex], we can express [tex]\( f(x) \)[/tex] as:
[tex]\[
f(x) = (x - 6)(x - 3)(x + 4)
\][/tex]
5. Verify the Factored Form:
To ensure that our factored form is correct, we can expand [tex]\( (x - 6)(x - 3)(x + 4) \)[/tex] and verify that it equals the original polynomial:
[tex]\[
(x - 6)(x - 3)(x + 4) = (x - 6)[(x - 3)(x + 4)]
\][/tex]
[tex]\[
(x - 3)(x + 4) = x^2 + x - 12
\][/tex]
[tex]\[
(x - 6)(x^2 + x - 12) = (x - 6)x^2 + (x - 6)x - 12(x - 6)
\][/tex]
[tex]\[
= x^3 - 6x^2 + x^2 - 6x - 12x + 72
\][/tex]
[tex]\[
= x^3 - 5x^2 - 18x + 72
\][/tex]
Thus, the polynomial [tex]\( x^3 - 5x^2 - 18x + 72 \)[/tex] can indeed be factored as [tex]\( (x - 6)(x - 3)(x + 4) \)[/tex].
So, the correct factorization is:
[tex]\[
\boxed{(x - 6)(x - 3)(x + 4)}
\][/tex]
