Basic Concepts Involved in Factoring Trinomials

Here, you will practice the basic concepts involved in factoring trinomials of the form $$x^{2}+bx+c$$.

These trinomials have an $$x^{2}$$ term with a coefficient of $$1$$, an $$x$$ term, and a constant term.

Recall that factoring is the process of taking a sum (things added) and rewriting it as a product (things multiplied).

Observe that for all real numbers $$f$$, $$g$$, and $$x$$:

$$(x+f)(x+g)=\overset{First}{\overbrace{x^{2}}}+\overset{Outer}{\overbrace{gx}}+\overset{Inner}{\overbrace{fx}}+\overset{Last}{\overbrace{fg}}=x^{2}+(f+g)x+fg$$

Now, think of going backwards:

from $$x^{2}+(f+g)x+fg$$ back to the factored form $$(x+f)(x+g)$$.

We would need two numbers that add together to give the coefficient of the $$x$$ term, and that multiply together to give the constant term.

This gives the following result, which is the primary tool used in factoring trinomials.

Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/basic_concepts.htm
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