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Chapter 0: Problem 55

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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 \((3x - 1)^{2}\).

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