Problem 87 - Introductory Algebra for College Students

Question

Factor each polynomial. $$24 x^{3} y^{3} z^{3}+30 x^{2} y^{2} z+18 x^{2} y z^{2}$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(24x^{3}y^{3}z^{3} + 30x^{2}y^{2}z + 18x^{2}y^{2}z^{2}\) is \(6x^{2}y^{2}z (4x^{2}y^{2}z^{2} + 5yz + 3z)\).

Step 1: Identify the GCF

Firstly, determine the highest common factor for both the coefficients and the variables of all terms. The coefficient factors for 24, 30, and 18 are \(2 \times 2 \times 2 \times 3\), \(2 \times 3 \times 5\) and \(2 \times 3 \times 3\) respectively. This gives us a common factor of 6. For the variables \(x^{3}y^{3}z^{3}\), \(x^{2}y^{2}z\) and \(x^{2}yz^{2}\) gives us a common factor of \(x^{2}yz\)

Step 2: Factor out the GCF

After identifying the GCF, we then divide each term of the polynomial by the GCF. This leaves us with a polynomial of smaller degree in the parenthesis, as given below: \[ 6x^{2}y^{2}z (4x^{2}y^{2}z^{2} + 5yz + 3z)\]

Step 3: Check your work

Multiply the factored form to verify that we get the original polynomial. Multiply the outside term by each of the terms within the parentheses: \(6x^{2}y^{2}z * 4x^{2}y^{2}z^{2} = 24x^{3}y^{3}z^{3}\), \(6x^{2}y^{2}z * 5yz = 30x^{2}y^{2}z\), and \(6x^{2}y^{2}z * 3z = 18x^{2}y^{2}z^{2}\). This verifies the factored form.