Problem 54 - Algebra and Trigonometry

Question

Factor each perfect square trinomial. $$25 x^{2}+10 x+1$$

Step-by-Step Solution

Answer

\((5x + 1)^2\)

Step 1: Identify the Terms

First, identify the terms of the trinomial. Here, \(a=5x\) because \((5x)^2=25x^{2}\), \(b=1\) because \(1^2=1\), and \(c=10x\) because \(2 * 5x * 1 = 10x\).

Step 2: Check if it's a Perfect Square Trinomial

Confirm whether the given trinomial is a perfect square trinomial or not. For a perfect square trinomial, the coefficient of the middle term is equal to twice the product of the coefficients of the first and last term. Therefore, here, since \(2 * a * b = 2 * (5x) * 1 = 10x\) which is equal to the middle term in our equation, we can say that this is a perfect square trinomial.

Step 3: Use the factoring formula

Since \(25x^2 + 10x + 1\) is a perfect square trinomial, use the factoring formula for perfect square trinomials, which is \((a+b)^2\). Hence, the trinomial \(25x^2 + 10x + 1\) can be written as \((5x + 1)^2\).