Question

which is a factor of $405x^4 - 5x^2$?
a. $6x + 5$

b. $5x + 2$

c. $3x - 1$

d. $9x - 1$

Explanation:

Step1: Factor out the GCF

First, factor out the greatest common factor (GCF) from \(405x^{4}-5x^{2}\). The GCF of \(405x^{4}\) and \(5x^{2}\) is \(5x^{2}\). So we have:
\(405x^{4}-5x^{2}=5x^{2}(81x^{2} - 1)\)

Step2: Apply difference of squares

Notice that \(81x^{2}-1\) is a difference of squares, which can be factored as \(a^{2}-b^{2}=(a + b)(a - b)\), where \(a = 9x\) and \(b = 1\). So:
\(81x^{2}-1=(9x + 1)(9x - 1)\)

Step3: Combine the factors

Putting it all together, the factored form of \(405x^{4}-5x^{2}\) is:
\(405x^{4}-5x^{2}=5x^{2}(9x + 1)(9x - 1)\)

From the factored form, we can see that \(9x - 1\) is a factor.

Answer:

D. \(9x - 1\)