Problem 29 - College Algebra Essentials

Question

Factor each trinomial, or state that the trinomial is prime. $$4 x^{2}+16 x+15$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(4x^2 + 16x + 15\) is \((2x + 3)(2x + 5)\).

Step 1: Identify the coefficients

Identify the coefficients and the constant in the trinomial. In the given trinomial \(4x^2 + 16x + 15\), \(a = 4\), \(b = 16\), and \(c = 15\).

Step 2: Find product of \(a\) and \(c\)

Calculate the product of \(a\) and \(c\), which gives \(4 \cdot 15 = 60\). Now, look for two numbers that multiply to 60 and add up to 16.

Step 3: Identify the factors

The two numbers that fulfill these requirements are 6 and 10 because \(6 \cdot 10 = 60\) and \(6 + 10 = 16\). These two numbers will be used to split the middle term.

Step 4: Rewrite the middle term and factor by grouping

Rewrite the trinomial \(4x^2 + 16x + 15\) as \(4x^2 + 6x + 10x + 15\). Now, group the terms to factor by grouping, so we have \(2x(2x + 3) + 5(2x + 3)\).

Step 5: Final factorization

Since both terms have a common factor of \(2x + 3\), factor it out, leaving the factored form of the trinomial as \((2x + 3)(2x + 5)\).