Question

Factor completely 4x4 − 24x3 + 36x2.

4x2(x − 3)2
4(x − 3)2
4x2(x + 3)(x − 3)
4x(x2 − 6x + 9)

Answer 1

To factor the expression \(4x^4 - 24x^3 + 36x^2\) completely, we can start by identifying the greatest common factor (GCF).

1. The GCF of the coefficients (4, -24, and 36) is 4.
2. The lowest power of \(x\) in all terms is \(x^2\).

So, we can factor out \(4x^2\):

\[
4x^2(x^2 - 6x + 9)
\]

Next, we will factor the quadratic \(x^2 - 6x + 9\). This quadratic can be factored as follows:

\[
x^2 - 6x + 9 = (x - 3)(x - 3) = (x - 3)^2
\]

Now we can substitute this back into our factored expression:

\[
4x^2(x^2 - 6x + 9) = 4x^2(x - 3)^2
\]

Thus, the complete factorization of the expression \(4x^4 - 24x^3 + 36x^2\) is:

\[
\boxed{4x^2(x - 3)^2}
\]