Question

Rozłóż wielomian na czynniki najniższych stopni.
-3x²+5x+2
-4x³-4x²+3x
3x⁵-4x⁴+4x³
(2x+3)(x+1)-(x-1)(3x-2)
-3(x-5)(x+5)+(x+8)(x-8)-1
(x²+x-6)(x²+x+5)
(9x²-12x+4)(3x²-4x+4)
(-4x²-8x+5)(x²-2x-15)
rozwiąż równanie.
3(x-1)(x+2)-2x(x-1)(x+3)=0
(1-2x)²-(1-2x)(x²+3)=0
(x-3)(2x+2)=(x-3)(x²-6)
x²(2x-3)(x+2)=x³-2x²
9x²-6x+1=0
-2x²+x+1=0
5x²-2x=-3
9x²=3x+2
3x²-5x=x²+2x-3
-5x²+5x-4=-x²-7x+5
x⁴+8x²=9
-x⁴+7x²=3(x⁴+1)
(5x-8)(4-3x)+x(4-x)=0
(2x-3)(x+5)=(5x+9)(6+3x)
x(2x+7)-5=(x+4)(5-x)
(3x-1)(2-x)-x(x+3)=2

Answer 1

1.
W(x) = -3x² + 5x +2
Δ = 25 -4*(-3)*2 = 25 + 24 = 49
√Δ = 7
x1= [-5 - 7]/(-6) = -12/(-6) = 2
lub x2 = [ -5 + 7]/(-6) = 2/(-6) = -1/3
zatem W(x) = -3*(x + 1/3 )*(x - 2)
2.
w(x) = -4x³ - 4x² + 3x = x( -4x² - 4x + 3) =
= -4*x *(x - 0,5 )*(x + 1,5)
bo
Δ = (-4)² -4*(-4)*3 = 16 + 48 = 64
√Δ = 8
x1 = [4 -8]/(-8) = -4/(-8) = 1/2 = 0,5
x2 = [4 + 8]/(-8) = 12 /(-8) = -1,5
3.
W(x) = 3x⁵ - 4x⁴ +4x³ = x³*(3x² - 4x + 4)
bo
Δ = (-4)² -4*3*4 = 16 - 48 < 0
4.
W(x)=(2x +3)(x+1) - (x-1)(3x -2) = 2x²+2x +3x +3 -(3x² -2x-3x+2) = -x² +10x +1 = ( x -5 -√26)(x -5 +√26)
bo
Δ = 10² -4*(-1)*1 = 100 +4 = 104 = 4*26
√Δ = 2√26
x1 = [-10 -2√26]/(-2) = 5 +√26
x2 = [ -10 +2√26]/(-2) = 5 - √26
5.
W(x) = -3(x-5)(x+5) + (x +8)(x-8) - 1 =-3(x² -25)+x² -64 -1 =
= -3x² + 75 +x² - 65 = -2x² + 75 = -2( x-2,5√6)(x +2,5 √6)
bo Δ = -4*(-2)*75 = 600 = 100*6
√Δ = 10√6
x1 = -10√6/(-4) = 2,5 √6 oraz x2 = 10 √6 /(-4) = -2,5 √6
6.
W(x) = (x² +x -6)(x² +x + 5) = x⁴ +x³ +5x² +x³ +x² +5x -6x²-6x -30
= x⁴ + 2x³ - x - 30 = (x -2)(x+3)(x² + x + 5)
7.
W(x) = (9x² -12x +4)(3x² - 4x +4) = 9(x+1)(x+1)(3x² -4x +4)
bo Δ1 = 144 - 4*9*4 = 0 ; x1,2 = 8/(-8) = -1
oraz Δ2 = 16 - 4*3*4 < 0
8.
W(x) = (-4x² - 8x +5)(x² -2x -15) =
= -4*(x -0,5)(x +2,5)(x +3)(x - 5)
bo
Δ1= 64 -4*(-4)*5 = 64 + 80 = 144
√Δ1 = 12
x1 = [ 8 - 12]/(-8) = -4/(-8) = 0,5
x2 = [ 8 + 12]/(-8) = 20/(-8) = -2,5
Δ2 = 4 -4*1*(-15) = 4 + 60 = 64
√Δ2 = 8
x3 = [ 2- 8]/2 = -6/2 = - 3
x4 = [ 2 + 8]/2 = 10/2 = 5
======================================================
1.
3(x-1)(x+2) -2x(x -1)(x +3) = 0
3(x-1){ x + 2 - 2x(x +3)] = 0
3(x -1)[x+1 - 2x² - 6x ] = 0
3(x -1)(-2x² -5x +1) = 0
Δ = 25 -4*(-2)*1 = 25 + 8 = 33
√Δ = √33
x = [5 - √33]/(-4) lub x = [ 5 + √33]/(-4)
x = -5/4 + √33/4 lub x = -5/4 - √33/4
zatem
x1 = -5/4 - √33/4
x2 = -5/4 + √33/4
x3 = 1
---------------------------
z.2
(1 -2x)² -(1 -2x)(x² +3) = 0
(1 -2x)[(1 -2x -(x² +3)] = 0
(1 -2x)(1 -2x -x² - 3) = 0
(1 -2x)(-x² -2x -2) = 0
1 - 2x = 0 ---> x = 1/2
Δ = 4 - 4*(-1)(-2) = 4 + 8 = 12 = 4*3
√Δ = 2√3
x = [2 -2√3]/(-2) = -1 +√3
lub x = [2 + 2√2]/(-2) = -1 - √2
mamy
x1 = -1 - √2
x2 = -1 + √2
x3 = 1/2
---------------------------------------------------
z.3
(x -3)(2x +2) = (x -3)(x² - 6)
(x -3)[2x +2 - (x² -6)] = 0
(x -3)(2x +2 - x² + 6) = 0
(x-3)(-x² +2x +8) = 0
x1 = 3
Δ = 4 - 4*(-1)*8 = 4 + 32 = 36
√Δ = 6
x2 = [-2 - 6]/(-2) = -8/(-2) = 4
x3 = [-2 +6]/(-2) = 4/(-2) = -2
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x,4
x²(2x -3)(x +2) = x³ -2x²
x²(2x -3)(x +2) = x² (x -2)
x²[(2x-3)(x +2) - (x-2)] = 0
x²(2x² +4x -3x -6 -x + 2) = 0
x²(2x² - 4) = 0
x²(√2x -2)(√2x +2 ) = 0
x² = 0 lub √2x =2 lub √2x = -2
x 1 = 0
x2 = √2
x3 = - √2
---------------------------------------------
itd.