Problem 31 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The factorized form of the trinomial \(9x^{2} - 9x + 2\) is \((3x - 1)(3x - 2)\).

Step 1: Identify the coefficients

The coefficients of the trinomial \(9x^{2} - 9x + 2\) are \(a = 9\), \(b = -9\), and \(c = 2\).

Step 2: Use the AC method

The AC method requires finding two numbers that multiply to give 'ac' and add to give 'b'. Here, \(ac = 9(2) = 18\) and \(b = -9\). The numbers -6 and -3 multiply to give 18 and add up to -9.

Step 3: Break down the middle term

With the two numbers found, -6 and -3, the trinomial is rearranged by breaking down the middle term: \(9x^{2} - 6x - 3x + 2\). The -6x and -3x terms are the '-bx' term split using the numbers -6 and -3.

Step 4: Factor by grouping

By grouping the first two and the last two terms: \((9x^{2} - 6x) - (3x - 2)\). Each group is factored: \(3x(3x - 2) - 1(3x - 2)\). By factoring out the common binomial, the final factorized form of the trinomial is \((3x - 1)(3x - 2)\).