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

polynomial.

5.6 Finding Rational Zeros5.6 Finding Rational Zeros

L1.2.1

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

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

-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

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

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

)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

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

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

-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

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

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

a

acbbx

2

42

QUADRATIC FORMULA

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

Example #24)

Find ALL the ZEROs of the polynomial function.

Doesn’t FCTPOLY…Now what?

22)( 23 xxxxf

)1)(2()( 2 xxxf

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

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..

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.