Question

If ∝,β,∫ are the roots of x³+qx+r=0 then ∝²+β²+∫²=

Answer 1

Answer:

a)`alpha+beta+gamma=0`<br> `alphabeta+betagamma+gammaalpha=q`<br> `alphabetagamma=-r`<br> `2(alpha+beta+gamma)=0=b`<br> `x^3-bx^2+cx-d=0`<br> `C=alphabeta+alphagamma+beta^2+betagamma+betaalpha+gamma^2+gammaalpha+alphabeta+alpha^2+alphabeta`<br> `c=(alpha+beta+gamma)^2+(alphabeta+gammabeta+alphagamma)`<br> `c=q`<br> `d=(alpha+beta)(beta+gamma)(gamma+alpha)`<br> `d=alphabeta+alphagamma+beta^2+betagamma)(alpha+gamma)`<br> `d=alphabetagamma+alphagamma^2+beta^2gamma+betagamma^2+alpha^2beta+alpha^2gamma+beta^2gamma+alphabetagamma`<br> `d=r`<br> `x^3+2x+r=0`<br> `b)b=alphabeta+betagamma+gammaalpha=q`<br> `c=alphabeta^2gamma+betagammaa^2alpha+alpha^2betagamma`<br> `=alphabetagamma(alpha+beta+gamma)=0`<br> `d=alpha^2beta^2gamma^2=r^2`<br> `x^3-qx^2+0x-r^2=0`<br> `x^3-qx^2-r^2=0`.

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Answer 2

Step-by-step explanation:

Alpha + Beta + Gama = 0

Alpha*Beta + Beta*Gama + Gama*Alpha = q

Alpha² + Beta² + Gama = (Alpha+Beta+Gama)² - 2(Al*Bet + Bet*Ga + Ga*A)

= (0)² - 2(q)

= -2q

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