4.2 Properties of Division

On Friday, we learned how to divide polynomials. When we divide two numbers, the answer is called the quotient. One way polynomials can be divided is by using long division. To set up the problem put the denominator on the left and the numerator to the right. Make sure to leave space for any degree of x that is missing.

Let’s look at another example.

In this example, there is a remainder. The final answer is:

We can also divide a polynomial by a linear number using synthetic division. Here’s the work used to solve the first example using synthetic division.

The top numbers are the coefficients of each power of x. The number on the left is the zero of the factor (x-1 -> x=1). To solve, move the first number down, multiply it by the number on the left, and add it to the next number. The answer is x²+x-6 with a remainder of 0.

Numbers that are divisible divide without a remainder.

We also learned the remainder theorem and the factor theorem.

Remainder Theorem- the remainder of ƒ(x) divided by x-c equal to ƒ(c)

Factor Theorem- If ƒ(c)=0 then ƒ(x) has a factor of x-c. This means that if ƒ(x) is divisible by x-c (has no remainder) it is a factor of ƒ(x).