Problem 26 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$3 x^{2}-2 x-5$$

Step-by-Step Solution

Answer

The factorized form of the trinomial \(3x^{2} - 2x - 5\) is \((3x - 5)(x + 1)\)

Step 1: Identify the Coefficients and Constant

The trinomial is \(3x^{2} - 2x - 5\). Here, the coefficient of \(x^{2}\) is 3, the coefficient of x is -2, and the constant term is -5.

Step 2: Find the Product of the Coefficients and Constant

The product of the coefficient of \(x^{2}\) (3) and the constant term (-5) is -15.

Step 3: Find Two Numbers that Multiply to -15 and Add to -2

The two numbers that meet these conditions are -5 and 3, because -5 times 3 equals -15, and -5 plus 3 equals -2.

Step 4: Rewrite the Trinomial Using these numbers

We can break down the term -2x using the numbers -5 and 3. This leads to: \(3x^{2} - 5x + 3x - 5\)

Step 5: Group and Factor

Now group the terms and factor by grouping: \(x(3x - 5) + 1(3x - 5)\) Now, take out the common binomial factor, which is \(3x - 5\), So the factored form of the trinomial is: \((3x - 5)(x + 1)\)