Problem 25 - College Algebra Essentials

Question

Factor each trinomial, or state that the trinomial is prime. $$3 x^{2}-25 x-28$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(3x^{2} - 25x - 28\) is \((3x - 28)(x + 1)\).

Step 1: Identify the coefficient and constant

To begin with, break down the given trinomial, \(3x^{2} - 25x - 28\). Observe that the coefficient of the \(x^{2}\) term is 3, the coefficient of the \(x\) term is -25, and the constant term is -28.

Step 2: Find two numbers

Find two numbers that add up to -25 (the coefficient of the middle term) and multiply to -84 (the product of the coefficients of the first and last term). The numbers that fit these conditions are -28 and 1.

Step 3: Rewrite the trinomial

Next, rewrite the trinomial with four terms using the found numbers, splitting the middle term: \(3x^{2} - 28x + x - 28\).

Step 4: Factor by grouping

Now, use factor by grouping. This involves factoring out the common factor for the first two terms and the common factor for the last two terms: \(x(3x - 28) + 1(3x - 28)\).

Step 5: Factor out the common binomial

Lastly, factor out the common binomial, \((3x - 28)\), from the two groups: \((3x - 28)(x + 1)\).