5.5 Apply the Remainder and Factor Theorems What you should learn: Goal Goal 1 1 Divide polynomials and relate the result to the remainder theorem and the factor theorem. 5.5 The Remainder and Factor Theorem 5.5 The Remainder and Factor Theorem a) using Long Division b) Synthetic Division Goal Goal 2 2 Factoring using the “Synthetic Method” Goal Goal 3 3 Finding the other ZERO’s when given one of them. A1.1.5
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. using Long Division Synthetic Division. Goal. 2. Factoring using the “Synthetic Method”. Goal. 3. - PowerPoint PPT Presentation
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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
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
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
- ( )
- ( )
- ( )
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
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
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
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
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
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
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)
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)
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
5.65.6 Finding Rational Zeros
What you should learn:GoalGoal 11 Find the rational zeros of a