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Chapter 6: Problem 37

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

Short Answer

Expert verified
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 by step solution

01

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 \(100 y^{2}\)
02

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 \(50y^{2}\).
03

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, \((100 y^{5} 梅 50y^{2}) - (50 y^{3} 梅 50y^{2}) + (100 y^{2} 梅 50y^{2})\) results in \((2 y^{3} - y + 2)\). The factorization of the polynomial is \(50 y^{2}(2 y^{3} - y + 2)\).

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Most popular questions from this chapter

Describe a strategy that can be used to factor polynomials.

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. It takes a great deal of practice to get good at factoring a wide variety of polynomials.

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$2 y^{3}+3 y^{2}-50 y-75$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$27 x^{2}+75$$

Factor completely. $$7 x^{4}+34 x^{2}-5$$
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