Questions: Factor completely: 2u^2y^4-2u^2.

Transcript text: Factor completely: \[ 2 u^{2} y^{4}-2 u^{2} \text {. } \]

Solution

Solution Steps

To factor the given expression completely, we first look for common factors in each term. We notice that \(2u^2\) is a common factor in both terms. After factoring out \(2u^2\), we simplify the expression inside the parentheses.

Step 1: Identify Common Factors

The given expression is \(2u^2y^4 - 2u^2\). We first identify the common factor in both terms, which is \(2u^2\).

Step 2: Factor Out the Common Factor

We factor out \(2u^2\) from the expression: \[ 2u^2(y^4 - 1) \]

Step 3: Recognize the Difference of Squares

The expression inside the parentheses, \(y^4 - 1\), is a difference of squares. It can be rewritten as: \[ (y^2)^2 - 1^2 \] This can be factored further using the difference of squares formula: \[ (y^2 - 1)(y^2 + 1) \]

Step 4: Factor the Difference of Squares Again

The term \(y^2 - 1\) is also a difference of squares: \[ (y - 1)(y + 1) \]

Step 5: Combine All Factors

Substituting back, the completely factored form of the original expression is: \[ 2u^2(y - 1)(y + 1)(y^2 + 1) \]