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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 \(3 x^{2}-25 x-28\) is \((x-7)*(3x+4)\).

Step by step solution

01

Determining the sign of the factors

Since the coefficient of \(x^2\) is positive (there's a + before \(3x^2)\), and the constant term is negative (there's a 鈥� before 28), it is known that the factors of the trinomial will be of different signs. That means, one factor will be positive and the other will be negative.
02

Finding the Factors

Now we need to find two numbers that multiply to -84 (product of a=3 and c=-28, that is, \(3*(-28)\)) and add up to -25 (coefficient b is -25). The pair of numbers that fits these conditions are -4 and 21. The equation becomes \(3x^2 -4x - 21x - 28\).
03

Grouping

After finding the factors, group the resulting terms: \(3x^2 -4x\) and \(- 21x - 28\).
04

Factoring the Groups

Factor each binomial group. When factoring out a common term, it results in the following: \(x*(3x-4) -7*(3x-4)\).
05

Final Factorization

Since the expressions in parentheses are the same, the trinomial can definitely be factored. The factored form of the trinomial is \((x-7)*(3x+4)\).

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