6.9: Summary of Key Concepts
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.