To factor the given perfect square trinomial [tex]\(16x^2 + 8x + 1\)[/tex], follow these steps:
1. Identify the coefficient of [tex]\(x^2\)[/tex], the coefficient of [tex]\(x\)[/tex], and the constant term.
For the trinomial [tex]\(16x^2 + 8x + 1\)[/tex]:
- The coefficient of [tex]\(x^2\)[/tex] is 16.
- The coefficient of [tex]\(x\)[/tex] is 8.
- The constant term is 1.
2. Determine the square root of the first and last terms.
- The square root of [tex]\(16x^2\)[/tex] is [tex]\(4x\)[/tex].
- The square root of 1 is 1.
3. Check if the middle term is twice the product of these square roots.
To confirm that it is a perfect square trinomial:
[tex]\( 2 \times (4x) \times 1 = 2 \times 4x \times 1 = 8x \)[/tex], which matches the middle term.
4. Write the trinomial as the square of a binomial.
Since [tex]\( 16x^2 + 8x + 1 \)[/tex] fits the form [tex]\(a^2 + 2ab + b^2\)[/tex], we can factor it as:
[tex]\[
(4x + 1)^2
\][/tex]
Therefore, the factorization of the trinomial [tex]\(16x^2 + 8x + 1\)[/tex] is:
[tex]\[
(4x + 1)^2
\][/tex]
