Question

25-26 pre-college math semester a exam (p
question 3
completely factor each expression:
b. $9x^2 - 16y^4$

Explanation:

Step1: Identify the difference of squares

The expression \(9x^2 - 16y^4\) can be written as \((3x)^2 - (4y^2)^2\), which is a difference of squares. The formula for factoring a difference of squares is \(a^2 - b^2=(a + b)(a - b)\).

Step2: Apply the difference of squares formula

Let \(a = 3x\) and \(b = 4y^2\). Then, using the formula \(a^2 - b^2=(a + b)(a - b)\), we get:
\((3x)^2 - (4y^2)^2=(3x + 4y^2)(3x - 4y^2)\)
Now, check if \(4y^2\) can be factored further. But \(4y^2=(2y)^2\), and \(3x - 4y^2\) and \(3x + 4y^2\) don't have common factors or further difference of squares (since \(3x\) is not a perfect square in terms of \(y\)). Wait, actually, \(16y^4=(4y^2)^2=(2y)^4\)? No, wait, \(16y^4=(4y^2)^2=(2y)^4\)? No, \(4y^2=(2y)^2\), so \(16y^4=(4y^2)^2=(2y)^4\)? Wait, no, \(16y^4=(4y^2)^2\), and \(4y^2=(2y)^2\), so \(16y^4=(2y)^4\)? Wait, no, \((2y)^4 = 16y^4\), but we have \(9x^2-16y^4=(3x)^2-(4y^2)^2=(3x + 4y^2)(3x - 4y^2)\). Now, \(4y^2=(2y)^2\), so \(3x - 4y^2\) and \(3x + 4y^2\) are both binomials, and neither \(3x + 4y^2\) nor \(3x - 4y^2\) can be factored further using real numbers (since they are not differences of squares themselves, as \(3x\) is not a perfect square in terms of \(y\)). Wait, but actually, \(16y^4=(4y^2)^2=(2y)^4\)? No, let's re - express:

Wait, \(9x^2=(3x)^2\) and \(16y^4=(4y^2)^2\), so first factoring as difference of squares:

\(9x^2 - 16y^4=(3x)^2-(4y^2)^2=(3x + 4y^2)(3x - 4y^2)\)

Now, check \(4y^2=(2y)^2\), but \(3x - 4y^2\) and \(3x + 4y^2\) are not differences of squares (because \(3x\) is not a perfect square with respect to \(y\)). So the complete factorization is \((3x + 4y^2)(3x - 4y^2)\)

Wait, but actually, \(16y^4=(4y^2)^2=(2y)^4\)? No, \(4y^2=(2y)^2\), so \(16y^4=(4y^2)^2=(2y)^4\)? No, \((2y)^4=16y^4\), but we have \(9x^2 - 16y^4=(3x)^2-(4y^2)^2=(3x + 4y^2)(3x - 4y^2)\). And \(4y^2=(2y)^2\), but \(3x - 4y^2\) and \(3x + 4y^2\) can't be factored further. So the final factored form is \((3x + 4y^2)(3x - 4y^2)\)

Answer:

\((3x + 4y^2)(3x - 4y^2)\)