Problem 123 - College Algebra

Question

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

Step-by-Step Solution

Answer

The process of factoring out the greatest common factor from a polynomial involves firstly, identifying the coefficients and variables of the polynomial, then, identifying the GCF for both coefficients and variables, factoring out the GCF from each term and finally checking the solution. For example, for a polynomial \( 12x^3y^2 + 16x^2y^3 \), the GCF is \( 4x^2y^2 \) and after factoring out, the polynomial becomes \( 4x^2y^2(3x + 4y) \).

Step 1: Identify the Polynomial and its terms

Consider a polynomial example, for instance, \( 12x^3y^2 + 16x^2y^3 \). It has two terms, namely \( 12x^3y^2 \) and \( 16x^2y^3 \).

Step 2: Identify the GCF of the Polynomial

Identify the GCF of the coefficients and the variables separately. Coefficients here are 12 and 16, the GCF of which is 4. For variables, take the lowest power for each variable present. Here, \( x^2 \) and \( y^2 \) are the GCF for variables in the terms.

Step 3: Factor out the GCF

Factor the GCF out of each term in the polynomial. This is done by dividing each term in the polynomial by the GCF found in Step 2. So our polynomial \( 12x^3y^2 + 16x^2y^3 \) factored out by its GCF \( 4x^2y^2 \) will be \( 4x^2y^2(3x + 4y) \). We performed \( 12x^3y^2 / 4x^2y^2 \) to obtain \( 3x \) and \( 16x^2y^3 / 4x^2y^2 \) to get \( 4y \). These terms left after dividing are inside the brackets.

Step 4: Check the Solution

To verify the factoring, distribute the factor i.e., multiply it back into the parentheses. Here, it will be \( 4x^2y^2 * 3x + 4x^2y^2 * 4y = 12x^3y^2 + 16x^2y^3 \) which is our initial polynomial, confirms that the factorization is correct.