Dividing Polynomial Expressions
By dividing polynomial expressions, we will be able to change the structure of polynomials to see factors that can help us obtain roots of that polynomial. We are going to talk about two kinds of division: long division and synthetic division. Long division of polynomials is similar to long division with numbers that you have done for many years. Synthetic division comes from the structure of long division and will be a more efficient way to divide certain types of polynomials.
The remainder in division will be particularly important going forward since a remainder of \(0\) will mean a factor divides an expression evenly. The examples below will work with how to deal with remainders other than \(0\) as well.
The video below will explain the process of long division and then a different worked out example is shown below that.
Example 1: Divide \(4x^4+20x^3-15x^2-35x+8\) by \(x^2+5x-2\).
By dividing polynomial expressions, we will be able to change the structure of polynomials to see factors that can help us obtain roots of that polynomial. We are going to talk about two kinds of division: long division and synthetic division. Long division of polynomials is similar to long division with numbers that you have done for many years. Synthetic division comes from the structure of long division and will be a more efficient way to divide certain types of polynomials.
The remainder in division will be particularly important going forward since a remainder of \(0\) will mean a factor divides an expression evenly. The examples below will work with how to deal with remainders other than \(0\) as well.
The video below will explain the process of long division and then a different worked out example is shown below that.
Example 1: Divide \(4x^4+20x^3-15x^2-35x+8\) by \(x^2+5x-2\).
Example 2: Divide \(6x^4+11x^3-50x^2+37x-17\) by \(3x-5\).
The video below will explain the connection between synthetic division and long division. Synthetic division can be used to divide polynomials by a polynomial in the form \(x+a\) (only an \(x\) raised to the first power and \(a\) is a constant). Although you can technically rewrite expressions such as \(\left(2x+3\right)\) as \(\left(x+\dfrac{3}{2}\right)\) and do synthetic division, it gets tricky with working with the quotient so we are only going to use synthetic division when the coefficient on \(x\) is \(1\). Otherwise, use long division. Also, know that you can use long division for anything, but synthetic division will often be more efficient as we move forward.
Example 3: Divide \(2x^4-15x^3+53x-28\) by \(x-7\).
Quick Check
Quick Check Solutions
- Divide \(8x^3-6x^2-47x-16\) by \(4x+7\).
- Divide \(4x^4+7x^3-12x^2+x-26\) by \(x+3\).
Quick Check Solutions