Question

which of the following shows the given polynomial factored completely?
$50x^5 - 18x$

a) $2x(5x + 3)(5x + 3)$

b) $2x(25x^4 + 9)$

c) $2x(5x^2 - 3)(5x^2 + 3)$

d) $2x(25x^3 + 9)$

Explanation:

Step1: Factor out GCF

First, find the greatest common factor (GCF) of \(50x^5\) and \(18x\). The GCF of 50 and 18 is 2, and the GCF of \(x^5\) and \(x\) is \(x\). So factor out \(2x\):
\(50x^5 - 18x = 2x(25x^4 - 9)\)

Step2: Apply difference of squares

Notice that \(25x^4 - 9\) is a difference of squares, since \(25x^4=(5x^2)^2\) and \(9 = 3^2\). The formula for difference of squares is \(a^2 - b^2=(a - b)(a + b)\). Here, \(a = 5x^2\) and \(b = 3\), so:
\(25x^4 - 9=(5x^2 - 3)(5x^2 + 3)\)

Step3: Combine factors

Substitute back into the expression from Step1:
\(50x^5 - 18x = 2x(5x^2 - 3)(5x^2 + 3)\)

Answer:

C. \(2x(5x^2 - 3)(5x^2 + 3)\)