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Chapter 0: Problem 5

Factor out the greatest common factor. $$9 x^{4}-18 x^{3}+27 x^{2}$$

Short Answer

Expert verified
The factored form of the given equation \(9x^{4} - 18x^{3} + 27x^{2}\) is \(9x^{2}(x^{2} - 2x + 3)\).

Step by step solution

01

Identify the terms

The given polynomial contains three terms, namely \(9x^{4}\), \(-18x^{3}\), and \(+27x^{2}\).
02

Calculate the greatest common factor (GCF)

The GCF of 9, -18, and +27 is 9. As for the power of the common variable x, we need to find the highest power that can divide each of the terms without leaving a remainder. Hence, the GCF for \(9x^{4}\), \(-18x^{3}\) and \(+27x^{2}\) would be \(9x^{2}\).
03

Divide each term by the GCF

Each term in the polynomial needs to be divided by the GCF. Thus, we get \((9x^{4}/9x^{2}) - (18x^{3}/9x^{2}) + (27x^{2}/9x^{2})\). This simplifies to \(x^{2} - 2x + 3\).
04

Write out the factored form

Replacing the original equation with the GCF factored out, we get \(9x^{2}(x^{2} - 2x + 3)\).

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