To factor the expression [tex]\(16m^4 - 25\)[/tex], follow these detailed steps:
1. Recognize the difference of squares:
Notice that [tex]\(16m^4 - 25\)[/tex] resembles a difference of squares. In general, the difference of squares can be written as:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
In this case, identify [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]. We can rewrite the expression as:
[tex]\[
(4m^2)^2 - 5^2
\][/tex]
2. Apply the difference of squares formula:
Now, apply the difference of squares formula:
[tex]\[
(4m^2)^2 - 5^2 = (4m^2 - 5)(4m^2 + 5)
\][/tex]
3. Resulting factors:
The expression [tex]\(16m^4 - 25\)[/tex] factors to:
[tex]\[
(4m^2 - 5)(4m^2 + 5)
\][/tex]
Thus, the factorization of [tex]\(16m^4 - 25\)[/tex] is:
[tex]\[
(4m^2 - 5)(4m^2 + 5)
\][/tex]
This form cannot be factored further over the set of real numbers, as [tex]\(4m^2 - 5\)[/tex] and [tex]\(4m^2 + 5\)[/tex] are both irreducible quadratic polynomials.
