Problem 88 - Introductory Algebra for College Students

Question

Factor each polynomial. $$16 x^{2} y^{2} z^{2}+32 x^{2} y z^{2}+24 x^{2} y z$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(16 x^{2} y^{2} z^{2}+32 x^{2} y z^{2}+24 x^{2} y z\) is \(8x^{2}y z(2y z + 4z + 3)\).

Step 1: Identify the common factors

Firstly, identify the greatest common factor (GCF) in each term of the polynomial. In \(16 x^{2} y^{2} z^{2}\), \(32 x^{2} y z^{2}\), and \(24 x^{2} y z\), the GCF is \(8x^{2}yz\).

Step 2: Divide each term by their GCF

The next step is to divide each term in the expression by the identified GCF. So the terms become \(2y^{1}z^{1}\), \(4 z^{1}\) and \(3\) respectively.

Step 3: Write the final factored form

Now that we have factored out the GCF and simplified each term, the final factored form of the polynomial is the GCF multiplied by the simplified expressions. Hence the factored form of the polynomial is \(8x^{2}y z(2y z + 4z + 3)\).