Problem 10 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}-8 x+16$$

Step-by-Step Solution

Answer

The factorization of the trinomial \(x^{2}-8 x+16\) is \((x-4)^2\).

Step 1: Identify the Square Roots

The square root of the first term \(x^{2}\) is \(x\). The square root of the third term \(16\) is \(4\).

Step 2: Check if the Trinomial is a Perfect Square

To determine if the trinomial is a perfect square, the second term should be twice the product of the square roots. So, \(2 \times x \times 4 = 8x\). As the second term in the trinomial is \(-8x\), we compare it with the multiplication result and we observe that it does indeed match, but with an opposite sign.

Step 3: Factor the Trinomial

Because this is a perfect square trinomial, it can be factored as \((x-4)^2\). We can write it as \(x^{2}-8 x+16 = (x-4)^2\).

Step 4: Check the Factorization Using FOIL

The FOIL method stands for First, Outer, Inner, and Last. It's a way for remembering how to multiply terms in a binomial. So, we apply FOIL to \((x-4)(x-4) = x^{2}-4x-4x+16 = x^{2}-8x+16\). The result matches the original trinomial, hence the factorization is correct.