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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