Problem 32 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$9 x^{2}+5 x-4$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(9x^2 + 5x - 4\) is \((9x - 4)(x + 1)\).

Step 1 Title

Set the trinomial. It is in the form of \(ax^2 + bx + c\), which we can identify as \(9x^2 + 5x - 4\).

Step 2: Identify coefficients

From the trinomial, a=9, b=5 and c=-4.

Step 3: Find two numbers

Find two numbers that multiply to (a*c) -36 and add up to b=5.

Step 4: Break middle term

The two numbers are 9 and -4. Therefore, break down the middle term: Re-write \(9x^2 + 5x - 4\) as \(9x^2 + 9x - 4x - 4\).

Step 5: Group and factor by GCF

Group the terms to become \(9x^2 + 9x + -4x - 4\) and factor by GCF within each binomial to get \(9x(x + 1) - 4(x + 1)\).

Step 6: Factor out common binomial

There is a common binomial \((x + 1)\) within both the expressions which can be factored out and we get \((9x - 4)(x + 1)\).

Step 7: Final form

Thus, the factored form of \(9x^2 + 5x - 4\) is \((9x - 4)(x + 1)\). If it was impossible to find terms that fulfill the criteria of step 3, then the trinomial would be considered prime, but that's not the case here.