6.9: Summary of Key Concepts

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Summary of Key Concepts

Factoring

Factoring is the process of determining the factors of some product. Factoring is the reverse of multiplication.

Greatest Common Factor

The greatest common factor of a polynomial is the factor that is common to every term of the polynomial and also is such that

The numerical coefficient is the largest number common to each term.
The variables possess the largest exponents that are common to all the variables.

Factoring a Monomial from a Polynomial

If \(A\) is the greatest common factor of \(Ax+Ay\), then

\(Ax + Ay = A(x+y)\)

Factoring by Grouping

We are alerted to the idea of factoring by grouping when the polynomial we are considering

Has no factor common to all terms.
Has an even number of terms.

\(\begin{aligned}
\underbrace{A x+A y}_{A \text{ is common}}+ \underbrace{B x+B y}_{B \text { is common }} &=\underbrace{A(x+y)+B(x+y)}_{x+y \text{ is common}} \\
&=(x+y)(A+B)
\end{aligned}\)

Special products

\(\begin{array}{flushleft}
a^2 - b^2&=&(a+b)(a-b)\\
a^2+2ab+b^2&=&(a+b)^2\\
a^2-2ab+b^2&=&(a-b)^2
\end{array}\)

Fundamental Rule of Factoring

Factor out all common monomials first.
Factor completely.

Factoring Trinomials

One method of factoring a trinomial is to list all the factor pairs of both of the first and last terms and then choose the combination that when multiplied and then added produces the middle term.