Problem 29 - Algebra and Trigonometry

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 \( (2x+5) (2x+3) \).

Step 1: Identify the values of a, b and c

From the provided trinomial, we find that \(a=4\), \(b=16\), and \(c=15\).

Step 2: Calculating the product of a and c

Next, multiply \(a\) and \(c\). In this case, \(4 * 15 = 60\). The two numbers we are looking for must multiply to give 60.

Step 3: Find two numbers that fulfill the requirement

The task is to find two numbers that add up to \(b\), which is 16, and multiply to the product calculated in step 2, which is 60. After some trial and error, we see that these numbers are 6 and 10, since \(6 * 10 = 60\) and \(6 + 10 = 16\).

Step 4: Rewrite the original expression

Now rewrite \(16x\) as \(6x + 10x\) in the original trinomial to get \(4x^2 + 6x + 10x + 15\).

Step 5: Group and factor

Group the terms and factor by grouping to get \(2x(2x + 3) + 5(2x + 3)\).

Step 6: Factored form

Finally, since both terms have the common factor \(2x + 3\), factor out \(2x + 3\) to obtain the final factored form \( (2x+5) (2x+3) \).