factor out the GCF (Greatest Common Factor), then rewrite the new, factored polynomial.
4x^4+8x^3+12x^2
To factor out the Greatest Common Factor (GCF) of the polynomial \(4x^4 + 8x^3 + 12x^2\), first we need to determine the GCF of the coefficients and the lowest power of \(x\) that appears in all terms.
The coefficients are 4, 8, and 12. The GCF of these numbers is 4 because 4 is the largest number that divides all of them evenly.
For the variable \(x\), we are looking at the terms \(x^4\), \(x^3\), and \(x^2\). The lowest power of \(x\) that appears in all terms is \(x^2\) because that is the highest power that can divide all three terms evenly.
Therefore, the GCF of the entire polynomial is \(4x^2\). To factor out the GCF, we can divide each term of the polynomial by \(4x^2\):
\[4x^4 ÷ 4x^2 = x^2\]
\[8x^3 ÷ 4x^2 = 2x\]
\[12x^2 ÷ 4x^2 = 3\]
So when we factor out the GCF \(4x^2\), the factored polynomial becomes:
\[4x^2(x^2 + 2x + 3)\]
And that's the factored form of the polynomial with the GCF factored out.