To factor the polynomial [tex]\( p(x) = x^3 + 7x^2 + 4x - 12 \)[/tex], we need to find its roots and express it as a product of its linear factors. Here are the steps to obtain the factored form:
1. Identify Possible Rational Roots:
By the Rational Root Theorem, any rational solution of the polynomial [tex]\( p(x) = x^3 + 7x^2 + 4x - 12 \)[/tex] must be a factor of the constant term (-12) divided by a factor of the leading coefficient (1). Thus, the possible rational roots are [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
2. Test for Roots:
We test these possible roots by substituting them into the polynomial to see which ones yield zero.
- [tex]\( p(1) = 1^3 + 7(1)^2 + 4(1) - 12 = 1 + 7 + 4 - 12 = 0 \)[/tex] (1 is a root)
Next, we use synthetic division or polynomial division to factor out [tex]\( (x - 1) \)[/tex] from [tex]\( p(x) \)[/tex].
3. Factor Polynomial:
Having found [tex]\( x = 1 \)[/tex] as a root, we divide [tex]\( p(x) \)[/tex] by [tex]\( (x - 1) \)[/tex] to get the quotient polynomial.
Performing synthetic division:
[tex]\[
\begin{array}{r|rrr}
1 & 1 & 7 & 4 & -12 \\
& & 1 & 8 & 12 \\
\hline
& 1 & 8 & 12 & 0 \\
\end{array}
\][/tex]
The quotient is [tex]\( x^2 + 8x + 12 \)[/tex], so:
[tex]\[
p(x) = (x - 1)(x^2 + 8x + 12)
\][/tex]
4. Factor the Quadratic Expression:
Next, we factor [tex]\( x^2 + 8x + 12 \)[/tex]:
We look for two numbers that multiply to 12 and add up to 8. These numbers are 2 and 6.
[tex]\[
x^2 + 8x + 12 = (x + 2)(x + 6)
\][/tex]
5. Write the Factored Form:
So, the factored form of [tex]\( p(x) = x^3 + 7x^2 + 4x - 12 \)[/tex] is:
[tex]\[
p(x) = (x - 1)(x + 2)(x + 6)
\][/tex]
Thus, the factored form of the polynomial [tex]\( p(x) = x^3 + 7x^2 + 4x - 12 \)[/tex] is [tex]\( (x - 1)(x + 2)(x + 6) \)[/tex].
