Answer :
Sure! Let's factor each polynomial completely, step by step.
Factor the polynomial [tex]25x^3 - 100x[/tex]:
First, identify the greatest common factor (GCF) of the terms. The terms are [tex]25x^3[/tex] and [tex]-100x[/tex]. The GCF is [tex]25x[/tex].
Factor out [tex]25x[/tex] from each term:
[tex]25x(x^2 - 4)[/tex]
Now, notice that [tex]x^2 - 4[/tex] is a difference of squares:
[tex]x^2 - 4 = (x - 2)(x + 2)[/tex]
So, the completely factored form is:
[tex]25x(x - 2)(x + 2)[/tex]
Factor the polynomial [tex]-16x + 36x^3[/tex]:
First, identify the GCF of the terms. The terms are [tex]-16x[/tex] and [tex]36x^3[/tex]. The GCF is [tex]4x[/tex].
Factor out [tex]4x[/tex]:
[tex]4x(9x^2 - 4)[/tex]
Now, notice that [tex]9x^2 - 4[/tex] is a difference of squares:
[tex]9x^2 - 4 = (3x - 2)(3x + 2)[/tex]
So, the completely factored form is:
[tex]4x(3x - 2)(3x + 2)[/tex]
Factor the polynomial [tex]-12x + 48x^3[/tex]:
First, identify the GCF of the terms. The terms are [tex]-12x[/tex] and [tex]48x^3[/tex]. The GCF is [tex]12x[/tex].
Factor out [tex]12x[/tex]:
[tex]12x(-1 + 4x^2)[/tex]
Now, recognize [tex]-1 + 4x^2[/tex] as another difference of squares, which can be rearranged and written as:
[tex]4x^2 - 1 = (2x - 1)(2x + 1)[/tex]
Therefore, the completely factored form is:
[tex]12x(2x - 1)(2x + 1)[/tex]
