5.5 apply the remainder and factor theorems what you should learn: goal1 divide polynomials and...
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5.55.5 Apply the Remainder and Factor Theorems
What you should learn:GoalGoal 11 Divide polynomials and relate the result to the
remainder theorem and the factor theorem.
5.5 The Remainder and Factor Theorem5.5 The Remainder and Factor Theorem
a) using Long Divisionb) Synthetic Division
GoalGoal 22 Factoring using the “Synthetic Method”
GoalGoal 33 Finding the other ZERO’s when given one of them.
A1.1.5
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Divide using the long division
ex)
3
23102
x
xx
23103 2 xxx
x
xx 32
x7 23
+ 7
217 x
2
)3(
2
x
- ( )
- ( )
6.5 The Remainder and Factor Theorem6.5 The Remainder and Factor Theorem
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Divide using the long division with Missing Terms
ex)
12
58 3
x
x
500812 23 xxxx23 48 xx
24x x0xx 24 2
52 x
24x x2 1
12 x
4
)12(
4
x
- ( )
- ( )
- ( )
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Synthetic DivisionTo 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
)3()554( 23 xxxx
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3. Write the leading coefficient of the dividend on the bottom row.
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
1 4 -5 5 3
1
1
3
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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 -5 5
1 4 -5 5 3
1
1
3
3
7
add
7
21 add
16
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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.
1 4 -5 5 3
1
3
7
add 21
16
48
53
5543 23 xxxx
3
531672
xxx
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Synthetic DivisionTo divide a polynomial by x - c
)1()24( 2 xxx
-1 1 4 -2
Example 1)
1
-1
3
-3
-5
1
53
xx
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Synthetic DivisionTo divide a polynomial by x - c
)2()75( 3 xxx
2 1 0 -5 7
Example 2)
1
2
2
4
-1
2
5122
xxx
-2
5
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Factoring a Polynomial
918112)( 23 xxxxfExample 1)
given that f(-3) = 0.
-3 2 11 18 9-6 -15 -9
2 5 3 0multiply
Because f(-3) = 0, you know that (x -(-3)) or (x + 3) is a factor of f(x).
918112 23 xxx )352)(3( 2 xxx
(x + 3)
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Factoring a Polynomial
1892)( 23 xxxxfExample 2)
given that f(2) = 0.
2 1 -2 -9 182 0 -18
1 0 -9 0multiply
Because f(2) = 0, you know that (x -(2)) or (x - 2) is a factor of f(x).
1892 23 xxx )9)(2( 2 xx
)3)(3)(2( xxx
(x - 2)
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Reflection on the SectionReflection on the SectionReflection on the SectionReflection 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)?
assignmentassignment
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5.65.6 Finding Rational Zeros
What you should learn:GoalGoal 11 Find the rational zeros of a
polynomial.
5.6 Finding Rational Zeros5.6 Finding Rational Zeros
L1.2.1
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Find the rational zeros of
5.6 Finding Rational Zeros5.6 Finding Rational Zeros
The Rational Zero Theorem
0
0
at coefficien leading offactor
a ermconstant tfactor
q
p
12112)( 23 xxxxf
solution List the possible rational zeros. The leading coefficient is 1 and the constant term is -12. So, the possible rational zeros are:
1
12,
1
6,
1
4,
1
3,
1
2,
1
1x
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Find the Rational Zeros of
5.6 Finding Rational Zeros5.6 Finding Rational Zeros
30772)( 23 xxxxf
solution
List the possible rational zeros. The leading coefficient is 2 and the constant term is 30. So, the possible rational zeros are:
30,15,10,6,5,3,2,1,2
15,
2
5,
2
3,
2
1x
Example 1)
Notice that we don’t write the same numbers twice
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-2 1 7 -4 -28
Example 2)
1
-2
5
-10
-14
)145)(2()( 2 xxxxf
28
0
Use Synthetic Division to decide which of the following are zeros of the function 1, -1, 2, -2
2847)( 23 xxxxf
)7)(2)(2()( xxxxfx = -2, 2
5.6 Finding Rational Zeros5.6 Finding Rational Zeros
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1 1 4 1 -6
Example 3)
1
1
5
5
6
)65)(1()( 2 xxxxf
6
0
Find all the REAL Zeros of the function.
64)( 23 xxxxf
)3)(2)(1()( xxxxfx = -2, -3, 1
5.6 Finding Rational Zeros5.6 Finding Rational Zeros
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2 1 1 1 -9 -10
Example 4)
1
2
3
6
7
14
5
Find all the Real Zeros of the function.
109)( 234 xxxxxf
10
0
-1 1 3 7 5
1
-1
2
-2
5
-5
0
5.6 Finding Rational Zeros5.6 Finding Rational Zeros
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)52)(1)(2()( 2 xxxxxf
x = 2, -1
-1 1 3 7 5
1
-1
2
-2
5
-5
0
5.6 Finding Rational Zeros5.6 Finding Rational Zeros
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Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
How can you use the graph of a polynomial function to help determine its real roots?
assignmentassignment
5.6 Finding Rational Zeros5.6 Finding Rational Zeros
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5.75.7 Apply the Fundamental Theorem of Algebra
What you should learn:GoalGoal 11 Use the fundamental theorem of
algebra to determine the number of zeros of a polynomial function.
5.7 Using the Fundamental Theorem of Algebra5.7 Using the Fundamental Theorem of Algebra
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
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-5 1 5 -9 -45
Example 1)
1
-5
0
0
-9
)9)(5()( 2 xxxf
45
0
Find all the ZEROs of the polynomial function.
4595)( 23 xxxxf
)3)(3)(5()( xxxxfx = -5, -3, 3
5.7 Using the Fundamental Theorem of Algebra5.7 Using the Fundamental Theorem of Algebra
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Example 1)
Decide whether the given x-value is a zero of the function.
55)( 23 xxxxf , x = -5
So, Yes the given x-valueis a zero of the function.
-5 1 5 1 5
1
-5
0
0
1
-5
0
5.7 Using the Fundamental Theorem of Algebra5.7 Using the Fundamental Theorem of Algebra
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Example 1)
Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1.
)5)(43()( 2 xxxxf
-4, 1, 5
20192)( 23 xxxxf
)5)(1)(4()( xxxxf
)5)(1)(4(0 xxx
5.7 Using the Fundamental Theorem of Algebra5.7 Using the Fundamental Theorem of Algebra
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a
acbbx
2
42
QUADRATIC FORMULA
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Example )
Find ALL the ZEROs of the polynomial function.
23)( 23 xxxf
)22)(1()( 2 xxxxf
)1(2
)2)(1(4)2()2( 2 x
x = 2.732 x = -.732
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Example #24)
Find ALL the ZEROs of the polynomial function.
Doesn’t FCTPOLY…Now what?
22)( 23 xxxxf
)1)(2()( 2 xxxf
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Example )
Find ALL the ZEROs of the polynomial function.
)1(2
)2)(1(4)2()2( 2 x
22)( 23 xxxxf
)161616)(1()( 23 xxxxxf
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Example )
)1495)(1()( 234 xxxxxxf
Find ALL the ZEROs of the polynomial function.
14131044)( 2345 xxxxxxf
-1 1 -4 4 10 -13 -14
1
-1
-5
5
9
-9
1
-1
-14
14
0
Graph this one….find one of the zeros..
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Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
How can you tell from the factored form of a polynomial function whether the function has a
repeated zero?
assignmentassignment
At least one of the factors will occur more than once.