Transcript Slide 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.
a) using Long Division
b) Synthetic Division
Goal 2
Factoring using the “Synthetic Method”
Goal 3
Finding the other ZERO’s when given one of them.
A1.1.5
5.5 The Remainder and Factor Theorem
Divide using the long division
ex)
x 2  10x  23
x3
2
x +7 
( x  3)
x  3 x  10x  23
2
- ( x  3x )
7 x  23
- ( 7 x  21 )
2
2
6.5 The Remainder and Factor Theorem
Divide using the long division with Missing Terms
ex)
8x3  5
2x 1
4x  2 x  1
3
2
2 x  1 8x  0 x  0 x  5
- (8x3  4 x 2 )
2
4x  0 x
- (4 x 2  2 x)
2
4

(2 x  1)
2x  5
- ( 2x 1 )
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.
( x3  4 x 2  5x  5)  ( x  3)
3 1
4
-5
5
3. Write the leading
coefficient of the dividend on
the bottom row.
3
1
4
-5
5
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
1
4
3
-5
5
5. Add the values in the
new column, writing
the sum in the bottom
row.
6. Repeat this series
of multiplications and
additions until all
columns are filled in.
3
1
4
add 3
1 7
3 1
1
-5
4
3
-5
21
7
16
5
5
add
7. Use the numbers in the
last row to write the
quotient and remainder in
fractional form.
3
1
4
add 3
1 7
The degree of the first
term of the quotient is one
less than the degree of the
first term of the dividend.
-5 5
21 48
16 53
53
x  7 x  16 
x 3
2
The final value in this row
is the remainder.
x  3 x  4 x  5x  5
3
2
Synthetic Division
To divide a polynomial by x - c
Example 1)
( x  4x  2)  ( x  1)
2
-1
1
1
4 -2
-1 -3
3
-5
5
x 3
x 1
Synthetic Division
To divide a polynomial by x - c
Example 2)
( x  5x  7)  ( x  2)
3
2
1
1
0 -5 7
2 4 -2
2 -1 5
5
x  2x 1
x2
2
Factoring a Polynomial
Example 1)
(x + 3)
f ( x)  2x  11x  18x  9
3
2
given that f(-3) = 0.
-3
2
11
-6
18
-15
9
-9
2
5
3
0
Because f(-3) = 0, you know that (x -(-3)) or (x + 3) is a factor of f(x).
2 x  11x  18x  9
3
2
 ( x  3)(2x  5x  3)
2
Factoring a Polynomial
Example 2)
(x - 2)
f ( x)  x  2 x  9 x  18
3
2
given that f(2) = 0.
2
1
-2
2
-9
0
18
-18
1
0
-9
0
Because f(2) = 0, you know that (x -(2)) or (x - 2) is a factor of f(x).
x  2 x  9 x  18
3
2
 ( x  2)(x  9)
2
 ( x  2)(x  3)(x  3)
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
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
The Rational Zero Theorem
factorconstant erm
t a0
p

q factorof leading coefficient a 0
Find the rational zeros of
f ( x)  x  2x 11x 12
3
2
solution List the possible rational zeros. The leading coefficient is 1
and the constant term is -12. So, the possible rational zeros
are:
1 2 3 4
6 12
x   , , , , ,
1 1 1 1
1
1
5.6 Finding Rational Zeros
Example 1)
Find the Rational Zeros of
f ( x)  2 x  7 x  7 x  30
3
2
solution
List the possible rational zeros. The leading coefficient is 2 and
the constant term is 30. So, the possible rational zeros are:
1
3
5 15
x   , , ,
,1,2,3,5,6,10,15,30
2
2
2
2
Notice that we don’t write the same numbers twice
5.6 Finding Rational Zeros
Use Synthetic Division to decide which of the following are zeros
of the function 1, -1, 2, -2
Example 2)
f ( x)  x  7 x  4x  28
3
-2
2
1
7 -4 -28
-2 -10 28
1 5 -14 0
f ( x)  ( x  2)(x  5x 14)
2
f ( x)  ( x  2)(x  2)(x  7)
x = -2, 2
5.6 Finding Rational Zeros
Find all the REAL Zeros of the function.
Example 3)
f ( x)  x  4 x  x  6
3
1
1
2
4
1
5
1
1
5
-6
6
6
0
f ( x)  ( x 1)(x  5x  6)
2
f ( x)  ( x  1)(x  2)(x  3)
x = -2, -3, 1
5.6 Finding Rational Zeros
Find all the Real Zeros of the function.
Example 4)
f ( x)  x  x  x  9 x 10
4
2
1
3
1
1
2
3
-1
1
1
2
1
6
-9 -10
14 10
7
3 7
-1 -2
2 5
5
0
5
-5
0
5.6 Finding Rational Zeros
-1
1
1
3 7
-1 -2
2 5
5
-5
0
f ( x)  ( x  2)(x  1)(x  2x  5)
2
x = 2, -1
5.6 Finding Rational Zeros
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
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 > 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
Find all the ZEROs of the polynomial function.
Example 1)
f ( x)  x  5x  9 x  45
3
-5
2
1
5 -9
-5 0
1 0 -9
-45
45
0
f ( x)  ( x  5)(x  9)
2
f ( x)  ( x  5)(x  3)(x  3)
x = -5, -3, 3
5.7 Using the Fundamental Theorem of Algebra
Decide whether the given x-value is a zero of the function.
f ( x)  x  5x  x  5 , x = -5
3
Example 1)
-5
2
1 5
-5
1 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
Write a polynomial function of least degree that has real
coefficients, the given zeros, and a leading coefficient of 1.
Example 1)
-4, 1, 5
0  ( x  4)(x  1)(x  5)
f ( x)  ( x  4)(x  1)(x  5)
f ( x)  ( x  3x  4)(x  5)
2
f ( x)  x  2x 19x  20
3
2
5.7 Using the Fundamental Theorem of Algebra
QUADRATIC FORMULA
 b  b  4ac
x
2a
2
Find ALL the ZEROs of the polynomial function.
Example )
f ( x)  x  3x  2
3
2
f ( x)  ( x 1)(x  2x  2)
2
 (2)  (2)  4(1)(2)
x
2(1)
2
x = 2.732
x = -.732
Find ALL the ZEROs of the polynomial function.
Example #24)
f ( x)  x  2x  x  2
3
2
f ( x)  ( x  2)(x  1)
2
Doesn’t FCTPOLY…Now what?
Find ALL the ZEROs of the polynomial function.
Example )
f ( x)  x  2x  x  2
3
2
f ( x)  ( x 1)(x  16x  16x  16)
3
2
 (2)  (2)  4(1)(2)
x
2(1)
2
Find ALL the ZEROs of the polynomial function.
Example )
-1
f ( x)  x  4x  4x  10x 13x 14
5
4
3
2
1 -4
-1
4
5
10
-9
-13
-1
-14
14
1 -5
9
1
-14
0
f ( x)  ( x 1)(x  5x  9x  x 14)
4
3
2
Graph this one….find one of the zeros..
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