Question

question
factor the polynomial expression $16y^4 - 625x^4$.
drag the expressions to the correct locations on the image. each expression can be used more than o expressions will be used.
$16y^4 - 625x^4 = (\quad)(\quad + \quad)(\quad - \quad)$
$4y^2 + 25x^2$ $4y^2$ $4y^2 - 25x^2$ $2y$ $5x$ $y$ $4y$ $25x$

Explanation:

Step1: Recognize as difference of squares

The expression \(16y^4 - 625x^4\) is a difference of squares since \(16y^4=(4y^2)^2\) and \(625x^4=(25x^2)^2\). Using the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\), we get:
\(16y^4 - 625x^4=(4y^2)^2-(25x^2)^2=(4y^2 + 25x^2)(4y^2 - 25x^2)\)

Step2: Factor \(4y^2 - 25x^2\) further

Notice that \(4y^2 - 25x^2\) is also a difference of squares, where \(4y^2=(2y)^2\) and \(25x^2=(5x)^2\). Applying the difference of squares formula again:
\(4y^2 - 25x^2=(2y)^2-(5x)^2=(2y + 5x)(2y - 5x)\)

Step3: Combine the factors

Substituting the factored form of \(4y^2 - 25x^2\) back into the previous result, we have:
\(16y^4 - 625x^4=(4y^2 + 25x^2)(2y + 5x)(2y - 5x)\)

Answer:

\(16y^4 - 625x^4 = (\boldsymbol{4y^2 + 25x^2})(\boldsymbol{2y} + \boldsymbol{5x})(\boldsymbol{2y} - \boldsymbol{5x})\) (or in the order of the blanks: first blank \(4y^2 + 25x^2\), second blank \(2y\), third blank \(5x\), fourth blank \(2y\), fifth blank \(5x\) when considering the structure \((\quad)(\quad+\quad)(\quad-\quad)\) as \((4y^2 + 25x^2)(2y + 5x)(2y - 5x)\))