To factor the perfect square trinomial [tex]\(16x^2 + 40x + 25\)[/tex], we follow several steps to identify its corresponding binomial squared.
1. Recognize the form of the trinomial:
The trinomial is in the form [tex]\(ax^2 + bx + c\)[/tex]. Here, [tex]\(a = 16\)[/tex], [tex]\(b = 40\)[/tex], and [tex]\(c = 25\)[/tex].
2. Identify the square roots of the first and last terms:
- The square root of [tex]\(16x^2\)[/tex] is [tex]\(4x\)[/tex].
- The square root of [tex]\(25\)[/tex] is [tex]\(5\)[/tex].
3. Verify that the middle term corresponds to [tex]\(2ab\)[/tex]:
For a trinomial to be a perfect square trinomial, it must fit the form [tex]\((ax + b)^2\)[/tex], which expands to [tex]\(a^2x^2 + 2abx + b^2\)[/tex]:
- Here, [tex]\(a = 4x\)[/tex] and [tex]\(b = 5\)[/tex].
- Compute [tex]\(2ab\)[/tex]: [tex]\(2 \times 4x \times 5 = 40x\)[/tex], which matches the middle term of our original trinomial.
Since all conditions are satisfied, we can factor the trinomial as:
[tex]\[
(4x + 5)^2
\][/tex]
Thus, the factorized form of [tex]\(16x^2 + 40x + 25\)[/tex] is:
[tex]\[
(4x + 5)^2
\][/tex]
