Polynomials

PolynomialsIf (x – a) is a factor of P(x), then P(a) = 0

PolynomialsIf (x – a) is a factor of P(x), then P(a) = 0

If (ax – b) is a factor of P(x), then = 0

PolynomialsIf (x – a) is a factor of P(x), then P(a) = 0

abPIf (ax – b) is a factor of P(x), then = 0

Multiple Roots

Multiple RootsIf P(x), has a root, x = a, of multiplicity m,

then P’(x) has a root, x = a, of multiplicity m - 1

Multiple RootsIf P(x), has a root, x = a, of multiplicity m,

then P’(x) has a root, x = a, of multiplicity m - 1

Proof: xQaxxP m

Multiple RootsIf P(x), has a root, x = a, of multiplicity m,

then P’(x) has a root, x = a, of multiplicity m - 1

Proof: xQaxxP m (m > 1, x = a is not a root of Q(x))

Multiple RootsIf P(x), has a root, x = a, of multiplicity m,

then P’(x) has a root, x = a, of multiplicity m - 1

Proof: xQaxxP m

11 mm axmxQxQaxxP

(m > 1, x = a is not a root of Q(x))

Multiple RootsIf P(x), has a root, x = a, of multiplicity m,

then P’(x) has a root, x = a, of multiplicity m - 1

Proof: xQaxxP m

11 mm axmxQxQaxxP

xmQxQaxax m 1

(m > 1, x = a is not a root of Q(x))

Multiple RootsIf P(x), has a root, x = a, of multiplicity m,

then P’(x) has a root, x = a, of multiplicity m - 1

Proof: xQaxxP m

11 mm axmxQxQaxxP

xmQxQaxax m 1

xRax m 1

(m > 1, x = a is not a root of Q(x))

Multiple RootsIf P(x), has a root, x = a, of multiplicity m,

then P’(x) has a root, x = a, of multiplicity m - 1

Proof: xQaxxP m

11 mm axmxQxQaxxP

xmQxQaxax m 1

xRax m 1

(m > 1, x = a is not a root of Q(x))

(where x = a is not a root of R(x))

Multiple RootsIf P(x), has a root, x = a, of multiplicity m,

then P’(x) has a root, x = a, of multiplicity m - 1

Proof: xQaxxP m

11 mm axmxQxQaxxP

xmQxQaxax m 1

xRax m 1

(m > 1, x = a is not a root of Q(x))

(where x = a is not a root of R(x))

P’(x) has a root, x = a, of multiplicity m - 1

e.g. (i) Solve the equation , given that it has a double root

01834 23 xxx

e.g. (i) Solve the equation , given that it has a double root

01834 23 xxx

1834 23 xxxxP

e.g. (i) Solve the equation , given that it has a double root

01834 23 xxx

1834 23 xxxxP 383 2 xxxP

e.g. (i) Solve the equation , given that it has a double root

01834 23 xxx

1834 23 xxxxP 383 2 xxxP

313 xx

e.g. (i) Solve the equation , given that it has a double root

01834 23 xxx

1834 23 xxxxP 383 2 xxxP

313 xx

double root is or 13

x 3x

e.g. (i) Solve the equation , given that it has a double root

01834 23 xxx

1834 23 xxxxP 383 2 xxxP

313 xx

double root is or

NOT POSSIBLEAs (3x + 1) is not a factor

13

x 3x

e.g. (i) Solve the equation , given that it has a double root

01834 23 xxx

1834 23 xxxxP 383 2 xxxP

313 xx

double root is or

NOT POSSIBLEAs (3x + 1) is not a factor

01834 23 xxx

13

x 3x

e.g. (i) Solve the equation , given that it has a double root

01834 23 xxx

1834 23 xxxxP 383 2 xxxP

313 xx

double root is or

NOT POSSIBLEAs (3x + 1) is not a factor

023 2 xx01834 23 xxx

13

x 3x

e.g. (i) Solve the equation , given that it has a double root

01834 23 xxx

1834 23 xxxxP 383 2 xxxP

313 xx

double root is or

NOT POSSIBLEAs (3x + 1) is not a factor

023 2 xx01834 23 xxx

3or 2 xx

13

x 3x

(ii) (1991)Let be a root of the quartic polynomial;

where

x 1234 AxBxAxxxP

22 42 AB

a) show that cannot be 0, 1 or -1

(ii) (1991)Let be a root of the quartic polynomial;

where

x 1234 AxBxAxxxP

22 42 AB

a) show that cannot be 0, 1 or -1 0 ,010 P

(ii) (1991)Let be a root of the quartic polynomial;

where

x 1234 AxBxAxxxP

22 42 AB

a) show that cannot be 0, 1 or -1 0 ,010 P

22

111

BA

ABAP

(ii) (1991)Let be a root of the quartic polynomial;

where

x 1234 AxBxAxxxP

22 42 AB

a) show that cannot be 0, 1 or -1 0 ,010 P

22

111

BA

ABAP 22

111

BA

ABAP

(ii) (1991)Let be a root of the quartic polynomial;

where

x 1234 AxBxAxxxP

22 42 AB

a) show that cannot be 0, 1 or -1 0 ,010 P

22

111

BA

ABAP 22

111

BA

ABAP

BUT 22 42 AB

(ii) (1991)Let be a root of the quartic polynomial;

where

x 1234 AxBxAxxxP

22 42 AB

a) show that cannot be 0, 1 or -1 0 ,010 P

22

111

BA

ABAP 22

111

BA

ABAP

BUT 22 42 AB

AB 22

(ii) (1991)Let be a root of the quartic polynomial;

where

x 1234 AxBxAxxxP

22 42 AB

a) show that cannot be 0, 1 or -1 0 ,010 P

22

111

BA

ABAP 22

111

BA

ABAP

BUT 22 42 AB

AB 22 022 BA

(ii) (1991)Let be a root of the quartic polynomial;

where

x 1234 AxBxAxxxP

22 42 AB

a) show that cannot be 0, 1 or -1 0 ,010 P

22

111

BA

ABAP 22

111

BA

ABAP

BUT 22 42 AB

AB 22 022 BA

01 ,01 PP

(ii) (1991)Let be a root of the quartic polynomial;

where

x 1234 AxBxAxxxP

22 42 AB

a) show that cannot be 0, 1 or -1 0 ,010 P

22

111

BA

ABAP 22

111

BA

ABAP

BUT 22 42 AB

AB 22 022 BA

01 ,01 PP1 hence

b) Show that is a root1

b) Show that is a root1

111234

ABAP

b) Show that is a root1

111234

ABAP

4

4321

ABA

b) Show that is a root1

111234

ABAP

4

4321

ABA

4P

b) Show that is a root1

111234

ABAP

4

4321

ABA

4P

40

b) Show that is a root1

111234

ABAP

4

4321

ABA

4P

40

root)aisas 0( P

b) Show that is a root1

111234

ABAP

4

4321

ABA

4P

40

root)aisas 0( P

0

b) Show that is a root1

111234

ABAP

4

4321

ABA

4P

40

root)aisas 0( P

0

xP ofroot a is 1

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

However P(x) is a quartic which has a maximum of 4 roots

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

However P(x) is a quartic which has a maximum of 4 roots

Thus no roots can have a multiplicity > 2

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

However P(x) is a quartic which has a maximum of 4 roots

Thus no roots can have a multiplicity > 2

ABxAxxxP 234 23

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

However P(x) is a quartic which has a maximum of 4 roots

Thus no roots can have a multiplicity > 2

ABxAxxxP 234 23

and 1 , be roots let the

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

However P(x) is a quartic which has a maximum of 4 roots

Thus no roots can have a multiplicity > 2

ABxAxxxP 234 23

and 1 , be roots let the

A431

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

However P(x) is a quartic which has a maximum of 4 roots

Thus no roots can have a multiplicity > 2

ABxAxxxP 234 23

and 1 , be roots let the

A431

)1(rootsofsum

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

However P(x) is a quartic which has a maximum of 4 roots

Thus no roots can have a multiplicity > 2

ABxAxxxP 234 23

and 1 , be roots let the

A431

)1(rootsofsum

B211

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

However P(x) is a quartic which has a maximum of 4 roots

Thus no roots can have a multiplicity > 2

ABxAxxxP 234 23

and 1 , be roots let the

A431

)1(rootsofsum

B211

)2(

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

However P(x) is a quartic which has a maximum of 4 roots

Thus no roots can have a multiplicity > 2

ABxAxxxP 234 23

and 1 , be roots let the

A431

)1(rootsofsum

B211

)2(

A41

c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB

roots 4for accounts which ,1 is so then , ofroot double a is If

xP

However P(x) is a quartic which has a maximum of 4 roots

Thus no roots can have a multiplicity > 2

ABxAxxxP 234 23

and 1 , be roots let the

A431

)1(rootsofsum

B211

)2(

A41

)3(

Substitute (3) into (1)

Substitute (3) into (1)

A

AA

211

43

411

Substitute (3) into (1)

A

AA

211

43

411

Substitute (3) into (2)

Substitute (3) into (1)

A

AA

211

43

411

Substitute (3) into (2)

BA

BA

BA

BAA

48 21

811

211

41-1

211

41

411

2

2

Substitute (3) into (1)

A

AA

211

43

411

Substitute (3) into (2)

BA

BA

BA

BAA

48 21

811

211

41-1

211

41

411

2

2

Exercise 5A; evens

Exercise 5B; 2, 4, 5b, 6b, 7b, 8 a,c,e,g,h

Substitute (3) into (1)

A

AA

211

43

411

Substitute (3) into (2)

BA

BA

BA

BAA

48 21

811

211

41-1

211

41

411

2

2

Exercise 5A; evens

Exercise 5B; 2, 4, 5b, 6b, 7b, 8 a,c,e,g,h

Note: tangent to a cubic has two solutions only.

Substitute (3) into (1)

A

AA

211

43

411

Substitute (3) into (2)

BA

BA

BA

BAA

48 21

811

211

41-1

211

41

411

2

2

Exercise 5A; evens

Exercise 5B; 2, 4, 5b, 6b, 7b, 8 a,c,e,g,h

Note: tangent to a cubic has two solutions only.

Substitute (3) into (1)

A

AA

211

43

411

Substitute (3) into (2)

BA

BA

BA

BAA

48 21

811

211

41-1

211

41

411

2

2

Exercise 5A; evens

Exercise 5B; 2, 4, 5b, 6b, 7b, 8 a,c,e,g,h

Note: tangent to a cubic has two solutions only. A double root

Substitute (3) into (1)

A

AA

211

43

411

Substitute (3) into (2)

BA

BA

BA

BAA

48 21

811

211

41-1

211

41

411

2

2

Exercise 5A; evens

Exercise 5B; 2, 4, 5b, 6b, 7b, 8 a,c,e,g,h

Note: tangent to a cubic has two solutions only. A double root

and a single root