Problem 83 - College Algebra Essentials

Question

Factor completely, or state that the polynomial is prime. $$20 y^{4}-45 y^{2}$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(20y^{4} - 45y^{2}\) is \(5y^{2}(2y + 3)(2y - 3)\).

Step 1: Factor out common terms

Firstly, identify the common terms in the given polynomial. In the polynomial \(20y^{4} - 45y^{2}\), 'y' and the number '5' can be seen as the common terms. Factor out these and rewrite the polynomial. This will give: \(5y^{2}(4y^{2} - 9)\).

Step 2: Check if remaining part can be factorized further

Now, consider whether the remaining part of the factorized polynomial, which is \(4y^{2} - 9\), can be factored further. This is a binomial and has the form \(a^{2} - b^{2}\), which is the difference of squares and can be factored as \((a+b)(a-b)\). Here, \(a = 2y\), \(b = 3\). Hence, the binomial \(4y^{2} - 9\) can be factored as \((2y + 3)(2y - 3)\).

Step 3: Write the completely factored form

Finally, replace \(4y^{2} - 9\) in your earlier factored form with its factored form to obtain the complete factored form of the original polynomial. This gives you \(5y^{2}(2y + 3)(2y - 3)\).