Problem 37 - 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. $$100 y^{5}-50 y^{3}+100 y^{2}$$

Step-by-Step Solution

Answer

The greatest common factor of the polynomial \(100 y^{5}-50 y^{3}+100 y^{2}\) is 50 and \(y^{2}\). When factored, the polynomial is expressed as \(50 y^{2}(2 y^{3} - y + 2)\).

Step 1: Identify the Terms in the Polynomial

First, identify the separate terms in the polynomial. Here, there are three terms: \(100 y^{5}\), \(-50 y^{3}\), and \(100 y^{2}\)

Step 2: Find the Greatest Common Factor

Next, find the greatest common factor (GCF) of the coefficient (numeric part) of all the terms and the variables. Here, the coefficients are 100, -50, and 100. The GCF of these numbers is 50. The variable for all the term is y with powers 5, 3 and 2. So, the common power of y to all terms is the smallest power which is 2. Therefore, the GCF of this polynomial is \(50y^{2}\).

Step 3: Factor Out the GCF

Factor out the GCF from each term in the polynomial. This is done by dividing each term by the GCF and then writing the result in parentheses after the GCF. So, \((100 y^{5} ÷ 50y^{2}) - (50 y^{3} ÷ 50y^{2}) + (100 y^{2} ÷ 50y^{2})\) results in \((2 y^{3} - y + 2)\). The factorization of the polynomial is \(50 y^{2}(2 y^{3} - y + 2)\).