Problem 123 - College Algebra Essentials

Question

Using an example, explain how to factor out the greatest common factor of a polynomial.

Step-by-Step Solution

Answer

Using the polynomial \(12x^3y^2 - 24x^2y^3 + 36x^3y\), we identify the GCF to be \(12x^2y\). Factoring out the GCF from each term in the polynomial gives us \(12x^2y(xy - 2y^2 + 3x)\).

Step 1: Identify the Polynomial

Let's use the example polynomial \(12x^3y^2 - 24x^2y^3 + 36x^3y\)

Step 2: Identify the GCF

Look at each term in the polynomial and identify the GCF. In this case, the GCF of \(12x^3y^2\), \(-24x^2y^3\), and \(36x^3y\) is \(12x^2y\). This is because each term can be divided evenly by \(12x^2y\).

Step 3: Factor out the GCF

Factor out the GCF from each term in the polynomial. This involves dividing each term in the polynomial by the GCF and writing the results as an expression that is multiplied by the GCF. So, \(12x^3y^2 ÷ 12x^2y = xy\), \(-24x^2y^3 ÷ 12x^2y = -2y^2\), and \(36x^3y ÷ 12x^2y = 3x\). Our factored polynomial is therefore \(12x^2y(xy - 2y^2 + 3x)\).