1. 5-3 Dividing Polynomials
Objectives
Students will be able to:
Divide polynomials using long division
Divide polynomials using synthetic division

2. When dividing a polynomial by a monomial, we break up the quotient into smaller quotients. We then divide the same way we divided monomials in section 5-1.
Look at how this expression is simplified.

3. Try and simplify this expression.

4. You can use a process similar to long division to divide a polynomial by a polynomial with more than one term.
Let’s quickly review long division.
Let’s apply a similar procedure now to dividing polynomials.

5. Steps for polynomial long division.
Rewrite the problem in long division form (if not already done).
In each polynomial, line the terms up in descending order, with respect to their degree.
Within the numerator (dividend), if a degree term is missing, add it in with a 0 coefficient.
Example: is missing the second degree term, so add it in.

6. 4) Begin dividing by taking the leading term of the numerator (dividend) and dividing it by the leading term of the denominator (divisor). Place the quotient on top of the term with the same degree in the dividend. 5) Multiply this quotient by all terms in the denominator (divisor) and place the result underneath the same degree terms of the numerator (dividend). 6) Subtract. 7) Bring down the next term of the dividend. Repeat the process.

7. Directions: Simplify each expression. 1)

8. Try this. 2)

9. 3)

10. 4)

11. Try these. 5) 6) 7)

12. Synthetic Division
Synthetic division: method of polynomial long division using only the coefficients of the terms
In order to use synthetic division, the denominator (divisor) must be a first degree binomial, which takes on the form
Also, the coefficient on the first degree term of the divisor must be a 1.
Let’s examine the steps of synthetic division while working through a problem.

13. First, let’s simplify this expression using long division
First, let’s simplify this expression using long division. Then we will compare the quotient to the quotient we attain using synthetic division.

14. Now let’s try using synthetic division.

15. Step 6: Write the quotient.

16. Let’s try simplifying another expression using synthetic division.

17. Try this one.