Long division can be used to divide polynomials in a similar way to dividing numbers. The key steps are to set up the division problem, divide the term of the dividend by the term of the divisor, multiply the divisor by the quotient term and subtract, then bring down the next term of the dividend and repeat. This polynomial long division allows polynomials to be factored by finding all divisor polynomials that give a remainder of zero. The factor theorem can also be used to check if a linear polynomial is a factor by setting it equal to zero and checking if it makes the other polynomial equal to zero.

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