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Chapter 0: Problem 25

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

Rewrite the trinomial

Next, rewrite the trinomial with four terms using the found numbers, splitting the middle term: \(3x^{2} - 28x + x - 28\).
04

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: \(x(3x - 28) + 1(3x - 28)\).
05

Factor out the common binomial

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

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