Factoring polynomials uses the same concept of factoring integers - we look for simpler monomials or binomials whose product is equal to the binomial/trinomial we’re factoring. Some techniques used in factoring polynomials include looking for common factors and using special factoring patterns.
Factor: : A number or term that is multiplied by another factor. To factor a number or polynomial is to find all of the factors for that number or polynomial.
Common Factor: A factor that appears in all terms of the polynomial. It can be a number, a variable, or a combination of numbers and variables.
Quadratic Polynomial: A polynomial of the 2nd degree.
Example: Factor \(12+4x\).
Recognizing these special products polynomials can make factoring easier.
Whenever we recognize these patterns, just figure out the values of \(a\) and \(b\) and we’re done!
A quadratic polynomial has the form \(ax^2 + bx + c\).
When \(a = 1\), the polynomial looks like \(x^2 + bx + c\).
When \(a = 1\), \(b > 0\), and \(c > 0\), both \(m\) and \(n\) are positive.
When \(a = 1\), \(b < 0\), and \(c > 0\), both \(m\) and \(n\) are negative.
When \(a = 1\) and \(c < 0\), either \(m\) or \(n\) (only one of them) will be negative.
When \(a = -1\), factor out the negative, then factor as usual.
A polynomial is factored completely when it can't be factored anymore.
Tips for factoring completely:
If there are four or more terms, sometimes we can only factor a common monomial from some of the terms. Go ahead and factor the common monomials from groups of terms, then see if there is a common binomial factor.
Example: Factor \(2x+2y+ax+ay\).
This can be used to solve quadratic expressions \(ax^2+bx+c\) where \(a ≠ 1\).
Example: Factor \(3x^2 + 8x + 4\).
It might be helpful to make an organized list when looking for two numbers that multiply to ac and add up tob. For the problem above, the list might look like this:
Zero-Product Property: If two numbers multiply to zero, then at least one of those numbers must be zero
If \(a ∙ b = 0\), then \(a = 0\) or \(b = 0\) (or both \(a\) and \(b\) equal \(0\)).
Factoring can be used to solve polynomial equations
Example: Solve \(x^2+7x = -6\).
It is very easy to make a sign error. Always check your answers by plugging them back into the equation!