ΠΟΛΥΩΝΥΜΑ ΑΛΓΕΒΡΑ Β ΛΥΚΕΙΟΥ
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ΠΟΛΥΩΝΥΜΑ 1. Αν α 3 +β 3 +γ 3 =3αβγ και α + β + γ 0, να δείξετε ότι το πολυώνυμο P(x)=(α-β)x 2 +(β-γ)x + γ - α είναι το μηδενικό πολυώνυμο. 2. Αν P(x) = x 15 – 4x 14 + 2x 3 + 2010, να υπολογιστεί η τιμή της παράστασης Α = 1 2 ⋅ [ P ( 2 + √ 2 )+ P( 2− √ 2 ) ] 3. Δίνεται το πολυώνυμο P(x) = (λ 2 – 4)x 4 + x 3 – 5x 2 + 6x +4λ + 6. Αν το πολυώνυμο έχει ρίζα το 1, να βρεθεί ο βαθμός του και οι άλλες του ρίζες. 4. Δίνεται ένα μη σταθερό πολυώνυμο P(x) και οι πραγματικοί αριθμοί α, β διάφοροι μεταξύ τους. Αν π 1 (x) και π 2 (x) τα πηλίκα των διαιρέσεων του πολυωνύμου P(x) με τα x–α και x – β αντίστοιχα, τότε: i . Να δείξετε ότι: π 1 (β) = π 2 (α) ii. Αν ισχύει: π 1 (x)P(x) = x π 2 3 (x) + π 1 2 ( x ) να βρείτε το πολυώνυμο Ρ(x) 5. Να λυθεί η εξίσωση √ x 2 + 8 x+7− √ x 2 +8 x=1 6. Να λυθεί η εξίσωση: (x 2 – 2x + 2) 3 + (x 2 – 2x – 1) + 1 = 0 7. Δίνονται τα πολυώνυμα P(x) = (x 2 – x – 1) 2004 (x 2 + x – 1) 2003 και Q(x) = (3x 2 – 5x + 1) 2005 (3x 2 – x – 1) 2006 . Να βρεθούν τα αθροίσματα των συντελεστών των πολυωνύμων P(x) ,Q(x) , Π(x) = P(x) Q(x)+2003 και T(x) = (P(x) – Q(x)) 11 – 44 8. Έστω το πολυώνυμο P(x) = 2x 3 – αx + β. Αν το υπόλοιπο της διαίρεσης του P(x) με το x 2 – 4 είναι 3x – 2 i. Να βρεθούν τα α και β ii. Να βρείτε το πηλίκο της διαίρεσής. iii. Να γράψετε την ταυτότητα της διαίρεσης. iv. Να βρείτε τα διαστήματα του x που η γραφική παράσταση του Ρ(x) βρίσκεται πάνω από την ευθεία ψ=3x – 2 . 9. Αν για το P(x) ισχύει 2 P(x) + P(2 – x) = – x 2 – 1 i. Να βρείτε τα P(0) και P(2) ii. Το υπόλοιπο της διαίρεσης του P(x) με το x 2 – 2x 1
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ΠΟΛΥΩΝΥΜΑ ΑΛΓΕΒΡΑ Β ΛΥΚΕΙΟΥ
Transcript of ΠΟΛΥΩΝΥΜΑ ΑΛΓΕΒΡΑ Β ΛΥΚΕΙΟΥ
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1. 3+3+3=3 + + 0, P(x)=(-)x2+(-)x + - .
2. P(x) = x15 4x14 + 2x3 + 2010, =
3. P(x) = (2 4)x4 + x3 5x2 + 6x +4 + 6. 1, .
4. P(x) , . 1(x) 2(x) P(x) x x , :i . : 1() = 2()
ii. : 1(x)P(x) = x (x) + (x)
5.
6. : (x2 2x + 2)3 + (x2 2x 1) + 1 = 0
7. P(x) = (x2 x 1)2004 (x2 + x 1)2003 Q(x) = (3x2 5x + 1)2005 (3x2 x 1)2006. P(x) ,Q(x) , (x) = P(x)Q(x)+2003 T(x) = (P(x) Q(x))11 44
8. P(x) = 2x3 x + . P(x) x2 4 3x 2 i. ii. .iii. .iv. x (x) =3x 2 .
9. P(x) 2 P(x) + P(2 x) = x2 1 i. P(0) P(2)ii. P(x) x2 2x
10.
P(x) = 8x3 15x2 + 4x 1 x 1 x - i. = = ii. P(x) > 0iii. (2 )
11. (x) 2 f(x) = (x 1) (x 2) : (x) = [ (2) (1) ] x + 2 (1) (2)
12. (x) 3 , (x2 + 1), 0 2.. (x) = x3 + x.. : (P(x) 2)3 + (P(x) 2)2 + P(x) > 2.
13. P(x) = x + -1x-1 + + 1x + P(x), P(P(P())) = 0 .
14. (x) , :
1) P(x+1) P(x) = x(x+1), x .2) P(0) = 0
, : 12 + 23 + 3 4 + + ( + 1)
15. P(x) = 4 x3 + 42x2 8x + 3.. P(x) Q(x) = x (, ).
16. P(x) Q(x) = x2 + x + 2009 .i) P(x) W(x) = x2 + x + 2010.ii) P(x) 20102 P(0)P(1).
17. Q(x) = x2 x + 1 P(x) = - 2x3 + x2 x +5 , .) . ) P(x):Q(x).) P(x) = 0.) (x) x'x .
18. (x) (0), (1) . .
19. (x) = 27x3 9x2 + 6x , , . = (x) , : || +2|| + || 1.
20. (x) = x x + - 1 *.i. (x) : (x 1)2ii. x 0: x + x +1.
21.
2(5x + 3) = 27 + 3+
22. (x) . (x) . (x) .
23. : (x 3m)(x m)(x + 2m)(x + 4m) = 2376m4 , m
24. f(x) = x3 x + g(x) = x3 x + . . i) .ii) : || + || 1iii) .
25. :) 2y3 3y + 1 = 0 ) 8x3 6x + = 0
26.
27.
28. :
29. x4 3x3 + 2x2 7x 1 = 0 : || < 13.
30. : x2 3x + 1 =
1 3+3+3 = 3 + + 0, P(x)=( )x2+( )x + .
Euler
3+3+3-3 = ( + + )[(-)2+(-)2+(-)2] = = .
2 P(x) = x15 4x14 + 2x3 + 2010,
=
= 2 + = 2 - . , x2 4x +2 = 0 x15 4x14 + 2x13 = 0 :
P() = 15 414 + 213 + 2010 = 0 + 2010 = 2010
P() = 15 414 + 213 + 2010 = 0 + 2010 = 2010 = = 2010
3 P(x) = (2 4)x4 + x3 5x2 + 6x +4 + 6. 1, .
x =1 P(1) = 02 4 + 1 5 + 6 + 4 + 6 = 0 2 + 4 + 4 = 0 ( + 2)2 = 0 = 2
= -2 : P(x) = x3 5x2 + 6x 2 3 .1-56-2=1
1-42
1-420
Horner
P(x)=(x-1)(P(x) =0 (x-1)(=0 (1) (1) ==16 8 = 8 > 0 x = 1, x =, x =
4 P(x) , . 1(x) 2(x) P(x) x x , :i) : 1() = 2()
ii) : 1(x)P(x) = x (x) + (x)
i) : P(x) = (x )1(x) + P() (1) P(x) = (x )2(x) + P() (2) x = (1) x = (2) : P() = ( )1() + P() P() = ( )2() + P() P() P() = ( )1() P() P() = ( )2() ( )1() = ( )2(), 1() = 2()
ii) (x) ( 1), 1(x) 2(x) 1.
x (x) : 1 + 3( 1) = 3 2
: 2( 1) = 2 2
3 2 > 2 2, x (x) + 2 2
1(x)P(x) 1 + = 2 1, 1(x)P(x) = x (x) + 2 1 = 3 2 = 1
(x) 1(x) = c1 2(x) = c2 c1, c2 .
:1() = 2() c1 = c2 = c 0 :
1(x)P(x) = x (x) + cP(x) = c3x + c2 P(x) = c2x + c
P(x) : (x ) c2 , o c2 = c c = 1 c 0, P(x) = x +1
5
x2 + 8x + 7 0 (2) x2 + 8x 0 (3) x2 + 8x + 7 = 0 :
=
(2) :
(3) :
= x2 + 8x :
- = 1
=+ 1
= (+ 1)2
+ 7 = +2+ 1
= 3 = 9
= 9 x2 + 8x = 9 x2 + 8x -9=0 x = 1 x = - 9 x = 1 x = - 9 (1) .
6 : (x2 2x + 2)3 + (x2 2x 1) + 1 = 0
R :
7 P(x) = (x2 x 1)2004 (x2 + x 1)2003 Q(x) = (3x2 5x + 1)2005 (3x2 x 1)2006. P(x) ,Q(x) , (x) = P(x)Q(x)+2003 T(x) = (P(x) Q(x))11 44
R(x) = x + -1x-1 + + 1x + , : R(1) = + -1+ + 1 + , :
8 P(x) = 2x3 x + . P(x) x2 4 3x 2 i. ii. .iii. .iv. x (x) =3x 2 .
i) (2) = 4 (2) = 8 = 5 = 2ii) (x) = 2xiii) (x) = (x2 4) 2x + 3x 2iv) Q(x) = 3x2P(x) Q(x) = 2x (x2 4) 2 < x < 0 x > 2
9 P(x) 2 P(x) + P(2 x) = x2 1 i. P(0) P(2)ii. P(x) x2 2x
i) x = 0 x = 2 :
(1) (2) P(0) = 1 , P(2) = 3
ii) P(x) = x ( x 2 ) Q(x) + x +
. = 1, = 2 P(x) 2x 2x +1
10
P(x) = 8x3 15x2 + 4x 1 x 1 x - i. = = ii. P(x) > 0iii. (2 )
) (1)= 0 () = 0. :
= , = .
) E P(x) = (x 1)2(2x 1) (x 1)2(2x 1) > 0 x (, 1) (1, +)
) 2 = (2 ) =
11 (x) 2 f(x) = (x 1) (x 2) : (x) = [ (2) (1) ] x + 2 (1) (2)
To P(x)=f(x)Q(x)+(x) (x) = x + .
P(1)= + (f(1)=0) P(2)=2+ (f(2)=0) = P(2) P(1) =2P(1) P(2)
(x) = [ (2) (1) ] x + 2 (1) (2)
12 (x) 3 , (x2 + 1), 0 2.. (x) = x3 + x.. : (P(x) 2)3 + (P(x) 2)2 + P(x) > 2.
) P(x) : P(x) = (x2 + 1)(x + ), 0 0, :(0)=0 (02 + 1)(0 + ) = 0.
P(1) = 2 ( (1) ).: 2 = 2 = 1.
:
P(x) = (x2 + 1)x = x3 + x, x
): G(x) = P(x) 2, x :G3(x) + G2(x) + G(x) > 0 G(x)(G2(x) + G(x) + 1) > 0
G(x) > 0 ( (G2(x) + G(x) + 1) > 0 x )
:G(x) > 0 P(x) 2 > 0 x3 + x 2 > 0 .(x 1)(x3 + x 2) > 0
x 1 > 0 x2 + x + 2 > 0 x x > 1
2
i) (x) (x2 + 1) (x)=(x2 + 1)(x) , (x) 1 (x) = x + , .
(0)=0 (0) = 0 = 0 (1) = 2 2(1) = 2 (1) = 1 + = 1 = 1.
(x) = x , (x) = x(x2 + 1) = x3 + x
ii) x, (x) 2 = , < 0.
(x) > 2
13 P(x) = x + -1x-1 + + 1x + P(x), P(P(P())) = 0 .
P(x), :
:
:
14 (x) , :
1) P(x+1) P(x) = x(x+1), x .2) P(0) = 0
, : 12 + 23 + 3 4 + + ( + 1)
i) P(x) = x3 + bx2 + cx + d .
:
P(0) = 0, :
: P(x) = x3 x .
ii) :
...!!!!
ii)
, . .
:
15 P(x) = 4 x3 + 42x2 8x + 3.. P(x) Q(x) = x (, ).
x , : () = 0. (x),:44 + 422 8 + 3 = 0 44 + 42(1 2) 8 + 3 = 0
42 8 + 3 = 0 ' : = () =()
= = x = 2+ x = 2+ , . (-, ) x = x = .
2
(x) x-,
, =
16 P(x) Q(x) = x2 + x + 2009 .i) P(x) W(x) = x2 + x + 2010.ii) P(x) 20102 P(0)P(1).
) G(x) ( ) . , , :
.
):(0)=2010G(0)(-1)=2010G(-1)
20102 P(0)P(1).( G(x), ).
17 Q(x) = x2 x + 1 P(x) = - 2x3 + x2 x +5 , .) . ) P(x):Q(x).) P(x) = 0.) (x) x'x .
i) :
ii)
iii)
iv)
18 (x) (0), (1) . .Problem-solving strategies by Engel
P(x) , :P(x) = (x )(x) (x) P(x) . x = 0 x = 1, :
(0) = (0) (1) = (1 ) (1)
(0), (1) .
(0), (1) , - , 1- . , 1 .
.
2 P(x) = x + -1x-1 + + 1x + + -1 + + 1 + . x p(x) = 2() + = .
x ( 2) P(x) 1 + -11-1 + + 11 +
1 + -11-1 + + 11 + = 1 (mod2) .
3 , (>=1, ), .
= 0 (0) . (x)=(x-)(x) () (1).
(1) x = 0 : (0)=(-)(0). (0) . .
(1) x = 1 (1)=(1-) (1). 1- , (1) . .
() .
19
(x) = 27x3 9x2 + 6x , , . = (x) , : || +2|| + || 1.
P(x), :()=0
:
20 (x) = x x + - 1 *.i. (x) : (x 1)2ii. x 0 : x + x +1.
i)
(0)
ii)
21
2(5x + 3) = 27 + 3+
, :
2 , x 1
= 3+
2 = 10x + 7 + 6= 10x + 6 +7
: = - 4 () = 5.
3+ = 5
x = 2 x =
22 (x) . (x) . (x) .
P(x) .
P(x)
P(x) = (x )(x) x =0 ,x = 1
( )
, (x) .
23
: (x 3m)(x m)(x + 2m)(x + 4m) = 2376m4 , m
:(x 3m)(x m)(x + 2m)(x + 4m) = 2376m4 (x 3m)( x + 4m)( x m)( x + 2m) = 2376m4 (x2 + mx 12m2)(x2 mx 2m2) = 2376m4.
: :
. : : . :x2 +mx 56m2 = 0 x2 + mx + 42m2 = 0.
: x = 7m x = -8m , .
24
f(x) = x3 x + g(x) = x3 x + . . i) .ii) : || + || 1iii) .
i) .
, :
ii)
iii) f(x) = 0 :
f(x) 0( x2 x + 1 )
g(x) :
+ .
25
:i) 2y3 3y + 1 = 0 ii) 8x3 6x + = 0
i) M Horner y = 1 .
ii) y = x i, :
2 ( ii )
x = y
3y = .
26
x < - 2 x > -1
: + = 2 = x = , x = x .
27
:
3y2 10y + 8 = 0 y = 2 y =
.
28
:
x - 5 x = 11, ... x > 11 5+x > 16 70+x > 81 :
> > 2 (1)
> > 3 (2) (1) (2) , :
+ > 5.
x > 11 .
, : , x = 11.
2
A = B = A + B = 5.
B4 A4 = 65 B :
2A3 15A2 + 50A 56 = 0, 2 .
A = 2 5 + x = 16 x = 11.
29 x4 3x3 + 2x2 7x 1 = 0 : || < 13.
|| 1, .
|| > 1, 4 33 + 22 7 1 = 0,
||3 > 0, :
, , || < 3 + 2 + 7 + 1 = 13
30
: x2 3x + 1 = R , :
x 0.
(3) :
(5) y = 2, y = 7 (4), y = 2, , x = 1, , (1).
y = 7, x = , , (1). x = 1.
