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Chapter 0: Problem 84

Factor completely, or state that the polynomial is prime. $$48 y^{4}-3 y^{2}$$

Short Answer

Expert verified
The completely factored form of the given polynomial is: \(3y^{2}(4y - 1)(4y + 1)\).

Step by step solution

01

Identify Common Factor

Look at the two terms and identify the common factor, if any. Here, both terms in the given polynomial, \(48y^{4}\) and \(3y^{2}\), have \(y^{2}\) as a factor and also both can be divided by the number 3.
02

Factor Out the Common Factor from Each Term

Factor out the common factor from each term. This means dividing each term by the common factor, and writing it once outside the bracket alongside the reduced expressions inside the bracket: \(3y^{2}(16y^{2} - 1)\).
03

Apply Difference of Squares Rule

The expression inside the bracket, \(16y^{2} - 1\), is in the form \(a^{2} - b^{2}\), which is a difference of squares. The difference of squares formula: \(a^{2} - b^{2} = (a - b)(a + b)\). Using this, \(16y^{2} - 1\) can be written as \((4y - 1)(4y + 1)\).
04

Write the Final Answer

Combine the factored part and the common factor to give the final factored form as: \(3y^{2}(4y - 1)(4y + 1)\).

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