Dividing Polynomials: Learn It 1

  • Divide polynomials using the long division method
  • Apply synthetic division as a quicker way to divide polynomials

With your skills in graphing polynomial functions, you can visualize how these functions behave. Now, let’s take it a step further by learning how to divide polynomials. This technique is crucial for breaking down complex polynomials and uncovering their zeros, which are essential for accurate graphing. Did you notice that all of the polynomials we graphed so far are in factored form? We get to this factored form through the process of dividing polynomials.

Dividing polynomials allows us to simplify expressions and find factors that reveal the zeros of the polynomial. These zeros, in turn, help us understand the roots and overall shape of the graph.

Polynomial Long Division

We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat.

For example, let’s divide [latex]178[/latex] by [latex]3[/latex] using long division.

Long Division. Step 1, 5 times 3 equals 15 and 17 minus 15 equals 2. Step 2: Bring down the 8. Step 3: 9 times 3 equals 27 and 28 minus 27 equals 1. Answer: 59 with a remainder of 1 or 59 and one-third.

 

Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.

[latex]\begin{array}{l}\left(\text{divisor }\cdot \text{ quotient}\right)\text{ + remainder}\text{ = dividend}\hfill \\ \left(3\cdot 59\right)+1 = 177+1 = 178\hfill \end{array}[/latex]

We call this the Division Algorithm and will discuss it more formally after looking at an another example.

Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials.

For example, if we were to divide [latex]2{x}^{3}-3{x}^{2}+4x+5[/latex] by [latex]x+2[/latex] using the long division algorithm, it would look like this:

Set up the division problem. 2x cubed divided by x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Then bring down the next term. Negative 7x squared divided by x is negative 7x. Multiply the sum of x and 2 by negative 7x. Subtract, then bring down the next term. 18x divided by x is 18. Multiply the sum of x and 2 by 18. Subtract.

 

We have found

[latex]\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\frac{31}{x+2}[/latex]

or

[latex]2{x}^{3}-3{x}^{2}+4x+5=\left(x+2\right)\left(2{x}^{2}-7x+18\right)-31[/latex]

We can identify the dividend, divisor, quotient, and remainder.

The dividend is 2x cubed minus 3x squared plus 4x plus 5. The divisor is x plus 2. The quotient is 2x squared minus 7x plus 18. The remainder is negative 31.

 

Writing the result in this manner illustrates the Division Algorithm.