1. Part 1: Long Division

2. Long Division
We can divide polynomials using steps that are
similar to the steps of numerical long division
a
Notation: a ÷ b = = b a
b
Vocabulary: dividend ÷ divisor = quotient

3. Example: Numerical Long Division
Divide using long division.
(Set up, Divide, Multiply, Subtract, Bring Down,
Repeat)
672 ÷ 21

4. Polynomial Long Division
Dividing polynomials is useful when we are trying to
factor polynomials, especially when we are unsure of
factors.

5. The Division Algorithm for Polynomials
An algorithm is a specific set of instructions used to
solve a problem.
The Division Algorithm for Polynomials is a
generalized version of the technique of long division
in arithmetic.
To divide polynomials, list polynomials in standard
form with zero coefficients where appropriate.

6. The Division Algorithm for Polynomials
You can divide a polynomial, P(x), by a polynomial,
D(x), to get a polynomial quotient, Q(x) and a
polynomial remainder, R(x).
Set up, Divide, Multiply, Subtract (change signs), Bring
Down, Repeat
Q( x)
D( x) P( x)
O
R( x)
The process stops when the degree of R(x) is less than
the degree of the divisor, D(x)

7. The Division Algorithm for Polynomials
The result is P(x) = D(x)Q(x) + R(x)
If there is no remainder, then D(x) and Q(x) are
factors of P(x)
To check your answers, multiply D(x) and Q(x) then
add R(x)

8. Example: Divide using long division. Check
your answers.
2 x +1 6 x + 7 x + 2
2

9. Example: Divide using long division. Check
your answers.
( 4x 2
+ 23 x − 16 ) ÷ ( x + 5)

10. Example: Divide using long division. Check
your answers.
( 3x − 29 x + 56 ) ÷ ( x − 7 )
2

11. Example: Divide using long division. Check
your answers.
(x 5
+ 1) ÷ ( x + 1)

12. Checking Factors
To check whether a polynomial is a factor of another
polynomial, divide.
If the remainder is zero, then the polynomial is a factor.

13. Example: Checking Factors
Is x
2
+ 1 a factor of 3 x 4 − 4 x 3 + 12 x 2 + 5 ?

14. Example: Checking Factors
Is x 4 − 1 a factor of x 5 + 5 x 4 − x − 5 ?

15. Checking Factors
If you need to check linear factors, we can use the
factor theorem.
Set the factor equal to zero and solve
Plug the value into the other polynomial and simplify
 If you get zero, then the factor you are checking is a factor of
the polynomial

16. Example: Checking Factors
Is x − 2 a factor of P ( x ) = x 5 − 32 ?
If it is, write P(x) as a product of two factors.

17. Homework
P308 #9 – 19 odd, 44 – 51 odd