Question

ZAD1 ROZŁÓŻ WIELOMIAN W NA CZYNNIKI

A) W(X)= 2X^5+2X^2

B) W(X) = 8X^4 -X

C) W(X)= -12X^5 +12X^3 -3X

D) W(X) = X^6 -18X^4 +81X^2

Answer 1

Wzory skróconego mnożenia:

(a+b)²=a²+2ab+b² - kwadrat sumy;

(a-b)²=a²-2ab+b² - kwadrat różnicy;

a²-b²=(a-b)(a+b) - różnica kwadratów;

(a+b)³=a³+3a²b+3ab²+b³ - sześcian sumy;

(a-b)³=a³-3a²b+3ab²-b³ - sześcian różnicy;

a³+b³=(a+b)(a²-ab+b²) - suma sześcianów;

a³-b³=(a-b)(a²+ab+b²) - różnica sześcianów;

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a)

W(X)= 2X^5+2X^2

W(x)=2x²(x³+1)

W(x)=2x²(x+1)(x²-x+1)

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b)

W(X) = 8X^4 -X

W(x)=8x(x³ - 1/8)

W(x)=8x(x - 1/2)(x² + x/2 +1/4)

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c)

W(X)= -12X^5 +12X^3 -3X

W(x)=-12x⁵+6x³+6x³-3x

W(x)=-6x³(2x²-1)+3x(2x²-1)

W(x)=(3x-6x³)(x² - 1/2)

W(x)=3x(1/2 - x²)(x² - 1/2)

W(x)=3x(√2/2 -x)(√2/2 +x)(x- √2/2)(x+ √2/2)

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D)

W(X) = X^6 -18X^4 +81X^2  

W(x)=x²(x⁴-9x²-9x²+81)

W(x)=x²[x²(x²-9)-9(x²-9)]

W(x)=x²(x-3)(x+3)(x-3)(x+3)

Answer 2

A) W(X)= 2X^5+2X^2=2X^2(X^3-1)=2X^2(X-1)(X^2+X+1)

B) W(X) = 8X^4 -X=X(8X^3-1)=X(2X-1)(4X^2+2X+1)

C) W(X)= -12X^5 +12X^3 -3X=-3X(4X^4-3X^2+1)

D) W(X) = X^6 -18X^4 +81X^2=X^2(X^4-18X^2+81)=X^2(X-3)(X-3)(X+3)(X+3)