Question

Rozłóż wielomian w na czynniki, grupujac jego wyrazy ; a) w(x)=x⁴+2x³-8x-16 B) w(x)=½x³-⅙x²-3x+1 c)w(x) =⅔x³-3x²-6x+27 d)w(x)=x³-√2x²+√2x-2 d)w(x)=x⁵+x⁴-2x³-2x²+x+1 Rozłóż wielomian w na czynniki a)w(x)=(20x³-28x²+8x)(x⁴+6x³+2x²+12x) b) w(x)=(-¼x⁴-2x³-4x²)(x³-7x²-4x+28) c)w(x)=(7x⁴+14x³-21x²)(x⁵-4x³-x²+4) d)w(x)=(3x⁴-2x³+⅓x²)(x⁶-1)

Answer 1

Rozłóż wielomian w na czynniki, grupujac jego wyrazy ;

a) w(x)=x⁴+2x³-8x-16= x³(x+2)-2(x+2)=(x+2)(x³-2)

B) w(x)=½x³-⅙x²-3x+1=½x²(x-1/3)-3(x-1/3)=(½x²-3)(x-1/3)

c)w(x) =⅔x³-3x²-6x+27=3x²(2/9x-1)-27(6/27x-1)=3x²(2/9x-1)-27(2/9x-1)=(3x²-27)(2/9x-1)=3(x²-27)(2/9x-1)=3(x-√27)(x+√27)(2/9x-1)=(x-√27)(x+√27)(2/3x-3)

d)w(x)=x³-√2x²+√2x-2=x²(x-√2)+√2(x-√2)=(x²+√2)(x+√2)

d)w(x)=x⁵+x⁴-2x³-2x²+x+1=x⁴(x+1)-2x²(x+1)+1(x+1)=(x+1)(x⁴-2x²+1)=(x+1)(x²-1)(x²-1)

 

Rozłóż wielomian w na czynniki

a)w(x)=(20x³-28x²+8x)(x⁴+6x³+2x²+12x)=4x(5x²-7x+2)(x³(x+6)+2x(x+6))=4x(5x²-7x+2)(x³+2x)(x+6))=4x(5x²-7x+2)x(x²+2)(x+6)=4x²(5x²-7x+2)(x²+2)(x+6)=

delta=49-4*5*2=49-40=9

√delta=3

x1=(7+3)/2*5=10/10=1

x2=(7-3)/2*5=4/10=0,4

=4x²*5(x-1)(x-0,4)(x²+2)(x+6)=20x²(x-1)(x-0,4)(x²+2)(x+6)

 

b) w(x)=(-¼x⁴-2x³-4x²)(x³-7x²-4x+28)=(x²(-¼x²-2x-4)(x²(x-7)-4(x-7))=(x²(-¼x²-2x-4)(x²-4)(x-7)=(x²(-¼x²-2x-4)(x-2)(x+2)(x-7)=-x²(¼x²+2x+4)(x-2)(x+2)(x-7)

delta=4-4*1/4*4=4-4=0

x1=-2/1/2=-2*2=-4

=-x²(¼(x+4)²)(x-2)(x+2)(x-7)=-¼x²(x+4)²)(x-2)(x+2)(x-7)

 

c)w(x)=(7x⁴+14x³-21x²)(x⁵-4x³-x²+4)=7x²(x²+2x-3)(x²(x³-1)-4(x³-1))=7x²(x²+2x-3)(x²-4)(x³-1)=7x²(x²+2x-3)(x-2)(x+2)(x-1)(x²+x+1)=

delta=4-4*1*(-3)=4+12=16

√delta=4

x1=(-2-4)/2=-6/2=-3

x2=(-2+4)/2=2/2=1

=7x²(x+3)(x-1)(x-2)(x+2)(x-1)(x²+x+1)=7x²(x+3)(x-1)²(x-2)(x+2)(x²+x+1)

 

 

d)w(x)=(3x⁴-2x³+⅓x²)(x⁶-1)=x²(3x²-2x+1/3)((x²)³-1³)=x²(3x²-2x+1/3)(x²-1)(x^4+x²+1)=x²(3x²-2x+1/3)(x-1)(x+1)(x^4+x²+1)=

delta=4-4*3*1/3=4-4=0

x1=2/6=1/3

=x²3(x-1/3)²(x-1)(x+1)(x^4+x²+1)=3x²(x-1/3)²(x-1)(x+1)(x^4+x²+1)