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)
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
1503222 2342 xxxxxx
2x2
+4x2-4x32x4-( )
- 4x27x3 +5x
7x3 = 7x x2
+7x
7x3 - 14x2 +14x-( )
10x2 - 9x -1
+10
10x2 - 20x +20-( )
11x - 21
remainder
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.
Multiply divisor by x3/x = x2.
x3 + 2x2
–3x2 + 4x Subtract. Bring down next term.Multiply divisor by –3x2/x = –3x.
– 3x2 – 6x
10x – 1 Subtract. Bring down next term.
Multiply divisor by 10x/x = 10.
10x + 20
– 30 remainder
x2 – 3x + 10 x + 2 x3 – x2 + 4x – 10)
quotient
Use Synthetic Division (x3 – x2 + 4x – 10) (x + 2) Set x + 2 = 0. Solve for x 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”
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)= 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
23
6
4
8
10
20
15
Which gives you: 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
2 1
2
4
8
2
4
-5
Which is x2 + 4x + 2 + -5 x-2
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.
211 18 9
-3 2
-6
5
-15
3
-9
0
(x + 3)(2x2 + 5x + 3)
So…. 2x3 + 11x2 + 18x + 9 factors to:
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) Write as a product of two factors.
= (x – 4)(x –3)(x + 1) Factor trinomial.
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.