Problem 34 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$15 x^{2}-19 x+6$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(15x^{2}-19x+6\) is \((3x-2)(5x-1)\).

Step 1: Identify the coefficients and constant

The coefficient of \(x^2\) is 15, the coefficient of \(x\) is -19, and the constant is 6. The goal is to find two numbers that multiply to \(15*6 = 90\) and add up to -19.

Step 2: Find the factors

The numbers that meet the criteria set in Step 1 are -10 and -9 because \(-10 * -9 = 90\) and \(-10 + -9 = -19\). Therefore, \(-10\) and \(-9\) are the needed numbers to factor the trinomial.

Step 3: Rewrite the trinomial as four terms

Express the middle term (-19x) as the sum of the terms -10x and -9x. The trinomial becomes \(15x^{2}-10x-9x+6\).

Step 4: Apply grouping

Group the four terms into two binomials. This results in \((15x^{2}-10x)-(9x-6)\).

Step 5: Factor out the greatest common factor (GCF)

Factor out the GCF from each binomial. The first binomial's GCF is \(5x\), and for the second binomial, it's -1. The trinomial now reads as \(5x(3x-2)-1(3x-2)\).

Step 6: Extract the common binomial

Both terms now have a common binomial factor of \(3x-2\). This allows the expression to be rewritten as \((3x-2)(5x-1)\).