Factor each perfect square trinomial. $64x^2-16x+1$

Verified step by step guidance

1. Recognize that the given trinomial \(64x^2 - 16x + 1\) is a perfect square trinomial, which generally has the form \(a^2 - 2ab + b^2\).
2. Identify the square terms: \$64x^2\( is \)(8x)^2\( and \(1\) is \)1^2$.
3. Check the middle term to confirm the pattern: the middle term should be \(-2 \times 8x \times 1 = -16x\), which matches the given middle term.
4. Write the trinomial as a square of a binomial: \((8x - 1)^2\).
5. Thus, the factorization of \(64x^2 - 16x + 1\) is \((8x - 1)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form a^2 ± 2ab + b^2, which factors into (a ± b)^2. Recognizing this pattern helps simplify factoring problems quickly.

Factoring Quadratic Expressions

Factoring quadratics involves rewriting the expression as a product of two binomials. For perfect square trinomials, this process is straightforward because the factors are identical binomials, making it easier to solve or simplify equations.

Identifying Squares and Middle Term

To factor a perfect square trinomial, identify the square terms (first and last) and verify if the middle term equals twice the product of their square roots. This check confirms the trinomial is a perfect square and guides the correct binomial factors.