Question

Factor the trinomial if possible. $$3 x^{2}-15 x+18$$

Short answer

The factored form of the trinomial \(3x^{2}-15x+18\) is \(3(x - 3)(x - 2)\).

Step-by-step solution

Identify coefficients

Identify the coefficients in the equation \(3x^{2}-15x+18\), where a = 3, b = -15, and c = 18.

Calculate ac product

Calculate the product of a and c, that is, 3 times 18 equals 54.

Factor pair of ac

Find the pair of numbers that multiply to give ac (product of 54) and add up to give b (-15). The numbers are -6 and -9.

Rewrite trinomial using factors

Rewrite the trinomial replacing the middle term (-15x) with the terms -6x and -9x; so \(3x^{2}-15x+18\) becomes \(3x^{2}-6x-9x+18\).

Group terms and factor by grouping

Group the terms as \( (3x^{2}-6x) - (9x-18) \), factor each group to get \(3x(x-2) - 9(x-2)\).

Common term

Take out the common term which is \((x-2)\) in this case to get \((3x-9)(x-2)\).

Simplify common terms

Simplify the terms in the binomial, \((3x - 9)\) becomes \(3(x - 3)\), which gives the final factored form of trinomial as \(3(x - 3)(x - 2)\).