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Polynomial Zeros Real Zeros of Polynomial Functions

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Page 1: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros

Real Zeros of

Polynomial Functions

Page 2: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 27/23/2013

Even/Odd Multiplicity Examples

Polynomial Functions

x

y(x)

x

y(x)

x

y(x)

x

y(x)

x

y(x)

●y = (x – 3)2

●y = x + 3

●y = (x – 3)3

●y = (x – 3)4

●y = (x – 3)5

●●

y = (x + 3)3(x – 3)

x

y(x)

y = (x + 2)3(x – 3)2

● ●

Page 3: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 37/23/2013

Rational Zero Test Let

f(x) = anxn + … + a2x2 + a1x + a0

for integer coefficients with an ≠ 0

Then all rational zeros of f(x) are of form

p/q p is a factor of a0 and q is a factor of an

p and q have no common factors

Polynomial Functions

Page 4: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 47/23/2013

Rational Zero Test

f(x) = anxn + … + a2x2 + a1x + a0

All rational zeros of f(x) of form p/q , with p is a factor of a0 and q is a factor of an

NOTE: This works only for integer coefficients NOT all zeros are rational numbers NO irrational zeros of f(x) are included

Polynomial Functions

Page 5: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 57/23/2013

Zeros and Factors FACT:

If two polynomials are equal then

they have the same factors

If f(x) = (x – k1)Q1(x) and if

Q1(x) = (x – k2)Q2(x) then we have

f(x) = (x – k1)(x – k2)Q2(x)

Polynomial Functions

Page 6: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 67/23/2013

Rational Zero Test Example Factor completely:

f(x) = 3x4 – 12x3 – 24x2 + 36x + 45

= 3(x4 – 4x3 – 8x2 + 12x + 15)

= 3g(x) Here an = a4 = 1 and a0 = 15 Factors p of 15 are: ±1, ±3, ±5, ±15 ; Factors q of 1 are: ±1 Possible p/q values are: ±1, ±3, ±5, ±15

Polynomial Functions

Page 7: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 77/23/2013

Rational Zero Test Example g(x) = x4 – 4x3 – 8x2 + 12x + 15

Possible p/q values: ±1, ±3, ±5, ±15 Check zeros of g(x)

Polynomial Functions

1 1 –4 –8 12 15

1

1

–3

–3

–11

–11

16 1

1k = 1

(x – 1) is NOT a factor of g(x)

1 –4 –8 12 15

1

3

–1

–3

–11

–33

–21

k = 3–63

–48 (x – 3) is NOT a factor of g(x)

3

Page 8: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 87/23/2013

Rational Zero Test Example Possible p/q values: ±1, ±3, ±5, ±15

Polynomial Functions

5 1 –4 –8 12 15

1

5

1

5

–3

–15

0 –3

–15 k = 5

(x – 5) IS a factor of g(x)

1 1 –3 –3

1

–1

0

0

–3

3

0

k = –1

(x + 1) IS a factor of Q1(x)

–1

Q1(x)

Q2(x)

Page 9: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 97/23/2013

Rational Zero Test Example

Polynomial Functions

Thus farf(x) = 3g(x) = 3(x – 5)Q1(x)

= 3(x – 5)(x + 1)Q2(x)= 3(x – 5)(x + 1)(x2 – 3)

Q2(x) = x2 – 3

with possible rational zeros of ±1 and ±3

Synthetic division shows none of these are zeros of Q2(x)

Page 10: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 107/23/2013

Rational Zero Test Example

Polynomial Functions

Q2(x) = x2 – 3

= 3 x2 – ( ) 2

= 3 (x – ) 3 (x + )

f(x) = 3(x – 5)(x + 1)Q2(x)Thus

3 (x – ) 3 (x + ) = 3(x – 5)(x + 1)

Two rational zeros and two irrational zeros

3 x = 5, –1, ±

Page 11: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 117/23/2013

Equations New functions f(x) lead to new types

of equations to solve Set f(x) = 0 and find the zeros

Examples

Find all real solutions of: 1. x4 – 1 = 0

Polynomial Functions

Page 12: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 127/23/2013

Equations Examples

Find all real solutions of:

2. x3 = x

3. x4 – 5x2 + 4 = 0

4. x6 – 19x3 = 216

Polynomial Functions

Page 13: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 137/23/2013

Problem Solving Find the width W of the rectangle from its length and area A. Also determine W when L = 5 inches.

A = WL

= W(x2 + 1)

= 3x3 + 3x – 5x2 – 5

= 3x(x2 + 1) – 5(x2 + 1)

= (3x – 5)(x2 + 1)

Polynomial Functions

W A = 3x3 – 5x2 + 3x – 5

L = x2 + 1

A

Page 14: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 147/23/2013

Problem SolvingA = (3x – 5)(x2 + 1)

Polynomial Functions

W A = 3x3 – 5x2 + 3x – 5

L = x2 + 1

W =x2 + 1

A

=x2 + 1

(3x – 5)(x2 + 1)

= 3x – 5

At x = 5 inches, W = 3(5) – 5 = 10 inches

Note: To find W we could have used conventional long division

Page 15: Polynomial Zeros Real Zeros of Polynomial Functions

Polynomial Zeros 157/23/2013

Think about it !