To determine which binomials are factors of the expression [tex]\(4x^2 - 25y^2z^2\)[/tex], let's carefully factor this expression step-by-step.
First, observe that the expression [tex]\(4x^2 - 25y^2z^2\)[/tex] resembles a difference of squares. The difference of squares is a standard algebraic identity given by:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
For the expression [tex]\(4x^2 - 25y^2z^2\)[/tex], we can rewrite it in the form [tex]\(a^2 - b^2\)[/tex]. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
[tex]\[
a^2 = 4x^2 \quad \text{and} \quad b^2 = 25y^2z^2
\][/tex]
From these equations, we find:
[tex]\[
a = 2x \quad \text{and} \quad b = 5yz
\][/tex]
Now apply the difference of squares formula:
[tex]\[
4x^2 - 25y^2z^2 = (2x)^2 - (5yz)^2 = (2x - 5yz)(2x + 5yz)
\][/tex]
Thus the expression [tex]\(4x^2 - 25y^2z^2\)[/tex] factors into:
[tex]\[
(2x - 5yz)(2x + 5yz)
\][/tex]
Therefore, the factors of the expression [tex]\(4x^2 - 25y^2z^2\)[/tex] are [tex]\(2x - 5yz\)[/tex] and [tex]\(2x + 5yz\)[/tex].
So, among the binomials provided, [tex]\(2x - 5yz\)[/tex] and [tex]\(2x + 5yz\)[/tex] are the factors of the expression [tex]\(4x^2 - 25y^2z^2\)[/tex].
