Today's soundtrack is Lil Wyte: Doubt Me Now, one of my favourite rap albums. An absolute classic.
This evening, I'm working on the first chapter of the third unit of my Pre-Calculus 11 course, "Factoring Polynomial Expressions."
The general form equation of a trinomial is ax² + bx + c.
Thus, if ever we are asked to find whether a given binomial is a factor of a given trinomial, we take that binomial out of the general equation and make it into a new equation, then we see if it equals the given trinomial.
For example, if we are asked to determine whether x + 3 is a factor of 2x² + x - 15, we would compare 2x² + x - 15 to (x + 3)(ax+b). We would start by using distributive properties on the second equation to find ax²_xb+3ax+3b, then we convert to trinomial form: ax² + (3a + b)x + 3b.
To factor a trinomial with rational coefficients, we multiply or divide as needed to ensure we are working with integer coefficients so that we end up with (x±y)(x±z).
To factor using a trinomial pattern, we use logical reasoning to find two numbers whose product is the last expression and whose sum is the middle expression so that we end up with (x±y)(x±z).
To factor using the difference of squares pattern, we identify a polynomial with the form [f(x)]²-[f(x)]², then factor it so that we end up with the form (x+y)(x-y).
According to Wyzant, factoring is a three-part process: "1. Factor a GCF from the expression, if possible. 2. Factor a Trinomial, if possible. 3. Factor a Difference Between Two Squares as many times as possible."
We know that an expression is fully factored once no further factoring is possible.