To factor the polynomial [tex]\(12x^4 - 6x^5 + 18x^3\)[/tex] with the greatest common factor, follow these steps:
1. Identify the common factor: First, identify the greatest common factor (GCF) of all the terms in the polynomial.
- The terms are [tex]\(12x^4\)[/tex], [tex]\(-6x^5\)[/tex], and [tex]\(18x^3\)[/tex].
- The numerical coefficients are 12, -6, and 18. The GCF of these coefficients is 6.
- Each term also contains a factor of [tex]\(x\)[/tex]. The smallest power of [tex]\(x\)[/tex] in these terms is [tex]\(x^3\)[/tex]. Therefore, the GCF also includes [tex]\(x^3\)[/tex].
- Thus, the GCF of the entire polynomial is [tex]\(6x^3\)[/tex].
2. Factor out the GCF: Divide each term of the polynomial by the GCF [tex]\(6x^3\)[/tex].
- For [tex]\(12x^4\)[/tex]:
[tex]\[ \frac{12x^4}{6x^3} = 2x \][/tex]
- For [tex]\(-6x^5\)[/tex]:
[tex]\[ \frac{-6x^5}{6x^3} = -x^2 \][/tex]
- For [tex]\(18x^3\)[/tex]:
[tex]\[ \frac{18x^3}{6x^3} = 3 \][/tex]
3. Rewrite the polynomial: After factoring out [tex]\(6x^3\)[/tex], you get:
[tex]\[ 12x^4 - 6x^5 + 18x^3 = 6x^3(2x - x^2 + 3) \][/tex]
4. Reorganize the expression in a common form: Let's rewrite the polynomial in a standard form with exponents in descending order:
[tex]\[
-6x^5 + 12x^4 + 18x^3 = -6x^3(x^2 - 2x - 3)
\][/tex]
5. Further factorization: Notice that you can factor the quadratic expression [tex]\(x^2 - 2x - 3\)[/tex] further:
- Solve the quadratic equation [tex]\(x^2 - 2x - 3 = 0\)[/tex].
- The roots of the equation are [tex]\(x = 3\)[/tex] and [tex]\(x = -1\)[/tex], so the quadratic can be factored as [tex]\((x - 3)(x + 1)\)[/tex].
6. Final Factorization: Substitute back these factors into the expression:
[tex]\[
-6x^5 + 12x^4 + 18x^3 = -6x^3(x - 3)(x + 1)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\(12x^4 - 6x^5 + 18x^3\)[/tex] is:
[tex]\[
-6x^5 + 12x^4 + 18x^3 = -6x^3(x - 3)(x + 1)
\][/tex]
