Algebra: Factoring Polynomials Part 1
Factoring Polynomials Part 1
This algebra video tutorial is the first part of two videos on factoring polynomials. The purpose of factoring will be explained in detail later as we apply the methodologies. Here, we will explains some basic techniques for factoring polynomials.
To factor a polynomial means to turn a polynomial sum into a product. For example, if we have the polynomial on the left-hand side of the equation below, we can use the distributive property to take out 2xy, as shown on the right side. This process of turning the polynomial sum on the left into the polynomial product on the right is called factoring. The polynomials 2xy and x - 5 are called factors of the left-hand side polynomial.
In the example above, we factored out 2xy from the polynomial. We could have factored out just 2 or x, but we always want to factor out as much as possible. So, we factored out 2xy.
We need to cover some terminology this is commonly used. Recall that the degree of a polynomial is equal highest degree of monomials in its sum. So, the three polynomials below have degrees 1, 2, and 3, respectively. These polynomials are called, linear, quadratic, and cubic, respectively based on their degrees. Additionally, a polynomial that has 1 as the coefficient of its highest degree term is called monic. So, the quadratic polynomial below is monic, while the other two are not.
One of the most common uses for factoring involves factoring a polynomial into linear polynomials. Often, we will have a moni polynomial that we can factor into monic linear polynomials. As an example of this, we have cubic polynomial below on the left side that is shown factor into a product of three linear polynomials on the right side of the equation.
The simplest and most common polynomial that we can factor into linear factors is a monic, quadratic polynomial. The formula for the general product of two linear polynomials is shown below, where a and b are constants. So, given a quadratic polynomial with integer coefficients, the first degree term is the sum of a and b, and the constant term is their product. If the coefficients of the quadratic are all integers then a and b are both integers. So, to factor a quadratic with integer coefficints, find all of the pairs of integers whose product is the constant term. Then check those to see whether any of the pairs sums to the first degree coefficient. If so, we have found our factorization.
To apply the method that I just discussed, begin with the quadratic polynomial below on the left side. Since this polynomial is not monic as it is, we factor out 3 to make it monic, as shown. After this, we can factor the inside monic quadratic with linear and constant terms -1 and -6, respectively. To begin, find all of the integer pairs whose product is -6. These are -1 and 6, -6 and 1, -3 and 2, and -2 and 3. These pairs sum to 5, -5, -1, and 1, respectively. Since we want them to sum to the linear term, -1, we choose a and b equal to -3 and 2. So, the monic quadratic factors as (x - 3)(x + 2).