Problem 33 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$20 x^{2}+27 x-8$$

Step-by-Step Solution

Answer

The factorized form of the trinomial \(20x^{2}+27x-8\) is \((x-0.25)(x+1.6)\).

Step 1: Identify the coefficients and constants

In the trinomial, \(20x^{2}+27x-8\), first identify the coefficient of \(x^{2}\) (here it's 20), the coefficient of \(x\) (here it's 27), and the constant term (here it's -8). The general form of the trinomial is \(ax^{2}+bx+c\).

Step 2: Factor the trinomial

In order to factorize the trinomial, we need to multiply \(a\) and \(c\), the coefficients of the \(x^{2}\) and the constant term respectively. So, multiply 20 and -8 to get -160. Now, we need to find two numbers that not only multiply to give -160 (the result), but also add up to give 27 (the coefficient of the \(x\) term). After breaking down -160 into pairs of factors, the numbers 32 and -5 are found. They add to 27 and multiply to -160.

Step 3: Rewriting the trinomial

Next, divide each term by the coefficient of \(x^{2}\) term. As a result, we rewrite the trinomial as \(x^{2}+1.35x -0.4\)

Step 4: Decompose the middle term and factor by grouping

Now, rewrite the middle term of the trinomial using the two numbers found in step 2: \(x^{2}+1.6x-0.25x-0.4\). Then group the terms and factor by grouping: \((x^{2}+1.6x) - (0.25x+0.4)\), which further simplifies to \(x(x+1.6) - 0.25(x+1.6)\). From here, one can see that the binomial \((x+1.6)\) is a common factor.

Step 5: Write final factorized form of the trinomial

We can thus write the trinomial in its final factorized form by displaying the common factor outside a new bracket, within which we write everything else as \((x-0.25)(x+1.6)\).