Apply the Remainder and Factor Theorems

1
5.5
Apply the Remainder and Factor Theorems
What you should learn
Goal
1
Divide polynomials and relate the result to the
remainder theorem and the factor theorem.

1. using Long Division
2. Synthetic Division

Goal
2
Factoring using the Synthetic Method
Goal
3
Finding the other ZEROs when given one of them.
A1.1.5
5.5 The Remainder and Factor Theorem
2
Divide using the long division
ex)
x
7
- ( )
- ( )
6.5 The Remainder and Factor Theorem
3
Divide using the long division with Missing Terms
ex)
- ( )
- ( )
- ( )
4
Synthetic Division To divide a polynomial by x - c
1. Arrange polynomials in descending powers, with
a 0 coefficient for any missing term.
2. Write c for the divisor, x c. To the
right, write the coefficients of the dividend.
3
1 4 -5 5
5
3
1 4 -5 5
3. Write the leading coefficient of the dividend
on the bottom row.
1
4. Multiply c (in this case, 3) times the value
just written on the bottom row. Write the
product in the next column in the 2nd row.
3
1 4 -5 5
3
1
6
5. Add the values in the new column, writing the
sum in the bottom row.
3
1 4 -5 5
3
add
1
7
6. Repeat this series of multiplications and
additions until all columns are filled in.
3
1 4 -5 5
21
3
add
16
7
1
7
7. Use the numbers in the last row to write the
quotient and remainder in fractional form. The
degree of the first term of the quotient is one
less than the degree of the first term of the
dividend. The final value in this row is the
remainder.
3
1 4 -5 5
48
3
21
add
1
7
16
53
8
Synthetic Division To divide a polynomial by x - c
Example 1)
-1
1 4 -2
-3
-1
1
3
-5
9
Synthetic Division To divide a polynomial by x - c
Example 2)
2
1 0 -5 7
-2
4
2
1
2
5
-1
10
Factoring a Polynomial
(x 3)
Example 1)
given that f(-3) 0.
2
11
18
9
-3
-6
-15
-9
2
5
3
0
multiply
Because f(-3) 0, you know that (x -(-3)) or (x
3) is a factor of f(x).
11
Factoring a Polynomial
(x - 2)
Example 2)
given that f(2) 0.
1
-2
-9
18
2
0
2
-18
1
0
-9
0
multiply
Because f(2) 0, you know that (x -(2)) or (x -
2) is a factor of f(x).
12
Reflection on the Section
If f(x) is a polynomial that has x a as a
factor, what do you know about the value of f(a)?
assignment
13
5.6
Finding Rational Zeros
What you should learn
Goal
1
Find the rational zeros of a polynomial.
L1.2.1
5.6 Finding Rational Zeros
14
The Rational Zero Theorem
Find the rational zeros of
solution
List the possible rational zeros. The leading
coefficient is 1 and the constant term is -12.
So, the possible rational zeros are
5.6 Finding Rational Zeros
15
Find the Rational Zeros of
Example 1)
solution
List the possible rational zeros. The leading
coefficient is 2 and the constant term is 30.
So, the possible rational zeros are
Notice that we dont write the same numbers twice
5.6 Finding Rational Zeros
16
Use Synthetic Division to decide which of the
following are zeros of the function 1, -1, 2, -2
Example 2)
-2
1 7 -4 -28
28
-10
-2
1
5
-14
0
x -2, 2
5.6 Finding Rational Zeros
17
Find all the REAL Zeros of the function.
Example 3)
1
1 4 1 -6
5
6
1
1
5
6
0
x -2, -3, 1
5.6 Finding Rational Zeros
18
Find all the Real Zeros of the function.
Example 4)
2
1 1 1 -9 -10
6
14
10
2
1
3
7
5
0
-1
1 3 7 5
-2
-5
-1
1
2
5
0
5.6 Finding Rational Zeros
19
-1
1 3 7 5
-2
-5
-1
1
2
5
0
x 2, -1
5.6 Finding Rational Zeros
20
Reflection on the Section
How can you use the graph of a polynomial
function to help determine its real roots?
assignment
5.6 Finding Rational Zeros
21
5.7
Apply the Fundamental Theorem of Algebra
What you should learn
Goal
1
Use the fundamental theorem of algebra to
determine the number of zeros of a polynomial
function.
THE FUNDEMENTAL THEOREM OF ALGEBRA
If f(x) is a polynomial of degree n where n gt
0, then the equation f(x) 0 has at least one
root in the set of complex numbers.
L2.1.6
5.7 Using the Fundamental Theorem of Algebra
22
Find all the ZEROs of the polynomial function.
Example 1)
-5
1 5 -9 -45
45
0
-5
1
0
-9
0
x -5, -3, 3
5.7 Using the Fundamental Theorem of Algebra
23
Decide whether the given x-value is a zero of the
function.
, x -5
Example 1)
-5
1 5 1 5
-5
0
-5
1
0
1
0
So, Yes the given x-value is a zero of the
function.
5.7 Using the Fundamental Theorem of Algebra
24
Write a polynomial function of least degree that
has real coefficients, the given zeros, and a
leading coefficient of 1.
-4, 1, 5
Example 1)
5.7 Using the Fundamental Theorem of Algebra
25
QUADRATIC FORMULA
26
Find ALL the ZEROs of the polynomial function.
Example )
x 2.732
x -.732
27
Find ALL the ZEROs of the polynomial function.
Example 24)
Doesnt FCTPOLYNow what?
28
Find ALL the ZEROs of the polynomial function.
Example )
29
Find ALL the ZEROs of the polynomial function.
Example )
-1
1 -4 4 10 -13 -14
-1
5
-9
-1
14
1
-5
9
1
-14
0
Graph this one.find one of the zeros..
30
Reflection on the Section
How can you tell from the factored form of a
polynomial function whether the function has a
repeated zero?
At least one of the factors will occur more than
once.
assignment