2.5 Apply the Remainder and
Factor Theorems p. 120
How do you divide polynomials?
What is the remainder theorem?
What is the difference between synthetic
substitution and synthetic division?
What is the factor theorem?
When you divide a Polynomial
f(x) by a divisor d(x), you get a
quotient polynomial q(x) with a
remainder r(x) written:
f(x) = q(x) + r(x)
d(x)
d(x)
The degree of the remainder
must be less than the degree
of the divisor!
Polynomial Long Division:
 You write the division problem in the
same format you would use for
numbers. If a term is missing in
standard form …fill it in with a 0
coefficient.
 Example:
 2x4 + 3x3 + 5x – 1 =

x2 – 2x + 2
2x2
x  2 x  2 2 x  3x  0 x  5 x  1
2
4
2x4 = 2x2
x2
3
2
2x2 +7x +10
x  2 x  2 2 x  3x  0 x  5 x  1
2
4
3
2
-( 2x4 -4x3 +4x2 )
7x3 - 4x2 +5x
-( 7x3 - 14x2 +14x )
7x3 = 7x
x2
10x2 - 9x -1
-( 10x2 - 20x +20 )
11x - 21
remainder
The answer is written:
 2x2 + 7x + 10 + 11x – 21
x2 – 2x + 2
 Quotient + Remainder over divisor
Now you try one!
 y4 + 2y2 – y + 5 =
y2 – y + 1
 Answer: y2 + y + 2 +
3
y2 – y + 1
2.
(x3 – x2 + 4x – 10)  (x + 2)
SOLUTION
Write polynomial division in the same
format you use when dividing numbers.
Include a “0” as the coefficient of x2 in the
dividend. At each stage, divide the term with
the highest power in what is left of the
dividend by the first term of the divisor. This
gives the next term of the quotient.
x2 – 3x + 10
x + 2 ) x3 – x2 + 4x – 10
x3 + 2x2
–3x2 + 4x
– 3x2 – 6x
10x – 1
10x + 20
– 30
quotient
Multiply divisor by
x3/x = x2.
Subtract. Bring
down next term.
Multiply divisor by –
3x2/x = –3x.
Subtract. Bring
down next term.
Multiply divisor by
10x/x = 10.
remainder
ANSWER
x3 – x2 +4x – 10
x+2
= (x2 – 3x +10)+
OR…
– 30
x+2
Use Synthetic Division
 (x3 – x2 + 4x – 10)
 (x + 2)
 Set x + 2 = 0.
x = −2
 Use − 2 as the divisor for synthetic
division which is the same as synthetic
substitution.
 Synthetic division can be used to divide
any polynomial by a divisor of the form
“x −k”
 Solve for x
Remainder Theorem:
 If a polynomial f(x) is divisible by (x – k),
then the remainder is r = f(k).
 Now you will use synthetic division (like
synthetic substitution)
 f(x)= 3x3 – 2x2 + 2x – 5
 Divide by x - 2
F(x) = x3 – x2 + 4x – 10  (x + 2)
SOLUTION
–2
ANSWER
1
−1
4
−10
–2
6
– 20
1 –3
10
– 30
f(x)= 3x3 – 2x2 + 2x – 5
Divide by x - 2
 Long division results in ?......
 3x2 + 4x + 10 +
15
x–2
 Synthetic Division:
 f(2) =
3
-2 2
-5
6
8
20
2
3
Which gives you:
4
10
15
3x2 + 4x + 10 + 15
x-2
Synthetic Division
 Divide x3 + 2x2 – 6x -9 by (a) x-2 (b) x+3
 (a) x-2

1
2
-6
-9
8
4
1
2
4
2
-5
2
Which is x2 + 4x + 2 + -5
x-2
Synthetic Division Practice cont.
 (b) x+3

1
-3
1
2
-6
-9
-3
3
9
-1
-3
0
x2 – x - 3
Factor Theorem:
 A polynomial f(x) has factor x-k if f(k)=0
 note that k is a ZERO of the function
because f(k)=0
Factoring a polynomial
 Factor f(x) = 2x3 + 11x2 + 18x + 9
 Given f(-3)=0
 Since f(-3)=0
 x-(-3) or x+3 is a factor
 So use synthetic division to find the
others!!
Factoring a polynomial cont.

2
 -3
2
11
18
-6
-15 -9
5
3
9
0
So…. 2x3 + 11x2 + 18x + 9 factors to:
(x + 3)(2x2 + 5x + 3)
Now keep factoring-- gives you:
(x+3)(2x+3)(x+1)
Your Turn…
Factor the polynomial completely given that
x – 4 is a factor.
f (x) = x3 – 6x2 + 5x + 12
SOLUTION
Because x – 4 is a factor of f (x), you know
that
f (4) = 0. Use synthetic division
to find the other factors.
4
1 –6
5
12
4
–8
–12
1 – 2
–3
0
Use the result to write f (x) as a product of
two factors and then factor completely.
f (x) = x3 – 6x2 + 5x + 12 Write original polynomial.
= (x – 4)(x2 – 2x – 3)
= (x – 4)(x –3)(x + 1)
Write as a product of two
factors.
Factor trinomial.
Your turn!
 Factor f(x)= 3x3 + 13x2 + 2x -8
 given f(-4)=0
 (x + 1)(3x – 2)(x + 4)
Finding the zeros of a polynomial
function
 f(x) = x3 – 2x2 – 9x +18.
 One zero of f(x) is x=2
 Find the others!
 Use synthetic div. to reduce the degree
of the polynomial function and factor
completely.
 (x-2)(x2-9) = (x-2)(x+3)(x-3)
 Therefore, the zeros are x=2,3,-3!!!
Your turn!
 f(x) = x3 + 6x2 + 3x -10
 X=-5 is one zero, find the others!
 The zeros are x=2,-1,-5
 Because the factors are (x-2)(x+1)(x+5)
 How do you divide polynomials?
By long division
 What is the remainder theorem?
If a polynomial f(x) is divisible by (x – k), then
the remainder is r = f(k).
 What is the difference between synthetic
substitution and synthetic division?
It is the same thing
 What is the factor theorem?
If there is no remainder, it is a factor.
Assignment
Page 124, 7, 9, 11-15 odd, 21-23
odd, 29-33 odd, 35- 37 all