Question

Rozłóż wielomanian na czynniki: a) x^4-5x^3 b) x^3-3x^2-28x c) x^5-4x^4-4x^3+16x^2 d) x^5-2x^4+5x^3 e) 4x^4+8x^3+4x^2 f) 2x^3+10x^2-18X-90

Answer 1

a) x⁴-5x³=

=x³(x-5)

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b) x³-3x²-28x=

=x(x²-3-28)=

=x(x+4)(x-7)

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x²-3x-28=(x+4)(x-7)

Δ=b²-4ac=(-3)²-4*1*(-28)=9+112=121

√Δ=11

x₁=[-b-√Δ]/2a=[3-11]/2=-4

x₂=[-b+√Δ]/2a=[3+11]/2=7

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c) x⁵-4x⁴-4x³+16x²=

=x²(x³-4x²-4x+16)=

=x²[x²(x-4)-4(x-4)]=

=x²(x²-4)(x-4)=

=x²(x+4)(x-4)(x-4)=

=x²(x+4)(x-4)²

[Wzór: a²-b²=(a+b)(a-b)]

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d) x⁵-2x⁴+5x³=

=x³(x²-2x+5)

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x²-2x+5  - brak pierwiastków

Δ=b²-4ac=(-2)²-4*1*5=4-20=-15<0

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e) 4x⁴+8x³+4x²=

=4x²(x²+2x+1)=

=4x²(x+1)²

[Wzór: (a+b)²=a²+2ab+b²]

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f) 2x³+10x²-18x-90=

=2x²(x+5)-18(x+5)=

=2(x²-9)(x+5)=

=2(x-3)(x+3)(x+5)