Problem 55 - College Algebra Essentials

Question

Factor each perfect square trinomial. $$9 x^{2}-6 x+1$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(9x^{2}-6x+1\) is \((3x - 1)^{2}\).

Step 1: Identify the perfect square terms

Observe that the first term \(9x^{2}\) is a perfect square of \(3x\), and the third term 1 is a perfect square of 1.

Step 2: Verify the middle term

The middle term should be twice the product of the square roots of the first and third terms. Therefore, multiply \(3x\) and 1 and double the result, which gives \(2 * 3x * 1 = 6x\). The middle term in the given trinomial is \(-6x\), confirming that the trinomial is indeed a perfect square.

Step 3: Factor the trinomial

Considering the observed conditions, the trinomial can be factored into the square of a binomial. The binomial has the square roots of the first and third terms of the trinomial, and the middle term sign. Therefore, the factored form of the trinomial \(9x^{2} - 6x + 1\) will be \((3x - 1)^{2}\).