Question

Factor the expression. $$48 y^{2}-72 x y+27 x^{2}$$

Short answer

The factored form of the given polynomial expression is \( (2 y-3 x)(24 y+9 x)\).

Step-by-step solution

Arrange the Polynomial Expression

To factor the expression by grouping, arrange the terms in the expression in descending order based on the terms' power. In this case, the given polynomial is already arranged correctly. Therefore, \(48 y^{2}-72 x y+27 x^{2}\) is the arranged polynomial.

Divide the Polynomial into Groups

Split the polynomial into 2 groups in a way that each group has common factors. So, the first group contains \(48 y^{2}-72 x y\) and the second group contains \(27 x^{2}\).

Factor Each Group

In each group, factor out the greatest common factor. For the first group, the greatest common factor is \(24 y\). Factoring that out yields \(2 y(24 y-3x)\). Note that there is no common factor in the second group, but each term in the first group was factored into expressions of the form \(a-bx\), so for consistency, doing the same for the second group. Therefore, \(27 x^{2}\) is factored as \(9 x(3x)\) .

Common Factor

Now there is a common factor in each of the expression terms. Factor out this expression \((2y-3x)\) from each term to obtain the final factored form , \( (2 y-3 x)(24 y+9 x)\).