Problem 45 - Introductory Algebra for College Students

Question

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state. $$30 x^{2} y^{3}-10 x y^{2}+20 x y$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(30x^{2}y^{3}-10xy^{2}+20xy\) is \(10xy(3xy^2 - y + 2)\).

Step 1: Identify the GCF

Identify the greatest common factor (GCF) for all terms. Since we are dealing with both coefficients and variables, we need to find the GCF for both. The GCF of the coefficients 30, -10, and 20 is 10. The variables in each term are \(x^{2} y^{3}\), \(xy^{2}\), and \(xy\). The GCF in the variables' part is \(xy\) since it appears in all terms.

Step 2: Divide each term by the GCF

Divide every term in the polynomial by the GCF. We will do this for both the coefficients and the variables: \[(30x^{2}y^{3} ÷ 10xy)= 3xy^{2}\] \[(-10xy^{2} ÷ 10xy) = -y\] \[(20xy ÷ 10xy) = 2\]

Step 3: Rewrite the polynomial

Rewrite the polynomial in factored form. Put the GCF out in front of the parentheses and write the results from step 2 inside the parentheses to obtain the factored form of the polynomial. The final answer should be \(10xy(3xy^2 - y + 2)\).