Computing factors of polynomials requires knowledge of different formulas
and some experience to find out which formula to be applied. Below, we give
some important formulas:
Given the expression , one may ask "what are the values of that make this expression 0?" If we factor we obtain
If , then one of the factors on the right becomes zero. Therefore, the whole must be zero. So, by factoring we have discovered the values of that render the expression zero. These values are termed "roots." In general, given a quadratic polynomial that factors as
then we have that and are roots of the original polynomial.
A special case to be on the look out for is the difference of two squares, . In this case, we are always able to factor as
For example, consider . On initial inspection we would see that both and are squares of and , respectively. Applying the previous rule we have
The AC method
There is a way of simplifying the process of factoring using the AC method. Suppose that a quadratic polynomial has a formula of
If there are numbers and that satisfy both
Then, we can rewrite the polynomial as
and factor out a common term from and to factor the polynomial.
For example, take the polynomial .
. and , so we can rewrite the polynomial as . From here, we can factor from the first two terms, and from the last two, which gives us . Thus, the factored form of is . Note that if we reverse the order of and , we get , which is the same factored form that we got previously. Thus, the order of and should not matter when using the AC method to factor a quadratic polynomial.
The quadratic formula
Given any quadratic equation , all solutions of the equation are given by the quadratic formula:
Note that the value of will affect the number of real solutions of the equation.
||There are two real solutions for the equation
||There are only one real solutions for the equation
||There are no real solutions for the equation
Remainder and Factor Theorem
The polynomial division algorithm is as follows: suppose and are nonzero polynomials where the degree of is greater than or equal to the degree of . Then there exist two unique polynomials, and , such that , where either or the degree of is strictly less than the degree of .
Suppose is a polynomial of degree at least 1 and c is a real number. When is divided by the remainder is .
- Proof: By the division algorithm, , where r must be a constant since has a degree of 1. must hold for all values of , so we can set and get that . Thus the remainder .
Suppose is a nonzero polynomial. The real number is a zero of if and only if is a factor of .
- By the division algorithm, is a factor of if and only if . So, since when is divided by , is a factor of if and only if ; that is, if is a zero of .
Factor Theorem Example Problem
- Determine if x + 2 is a factor of .
Since c is positive instead of negative we need to use this basic identity:
Now we can use the factor theorem.
Since the resultant is 0, is a factor of .
This means it is possible to re-state the polynomial in the form (x+2)( some linear expression of x).
Expanding the right hand side we get :
Equating like terms we get :
Giving , from the first and third equations and this works in the second, so
EXAMPLE 1: Find all the roots of
- Finding the roots is equivalent to solving the equation . Applying the quadratic formula with , we have:
- The quadratic formula can also help with factoring, as the next example demonstrates.
EXAMPLE 2: Factor the polynomial
- We already know from the previous example that the polynomial has roots and . Our factorization will take the form
- All we have to do is set this expression equal to our polynomial and solve for the unknown constant C:
- You can see that solves the equation. So the factorization is
EXAMPLE 3: Factor the polynomial .
- For this problem, let's use the AC method. , , and . So, , and we need to find two numbers whose product equals 12, and whose sum equals 7. Both ac and b are positive, so we are only concerned with the positive factors of 12, which are and . and , so we can rewrite the polynomial as .
Content obtained and/or adapted from: