Special question of sum and product of roots

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Q. (1). If 1,2,3 are the roots of the equation x³+ax²+bx+c=0 then what is the value of c?

Ans. Apply product of roots αβγ= -c/a=- constant term/coefficient of x³

                           1x2x3 =   -c/1 →   6 = -c →   c= - 6

Q.2. If (x+√2) is a factor of kx²-√2 x + 1, then what is the value of k?

Ans. Putting value of x in given equation so

   x+√2=0 then x= -√2 →   k (-√2)²- √2× (-√2) +1 = 0

                                                 2k+2+1=0

                                                                  2k+3=0 →   2k=-3   → k= -3/2

Q.3. What is the factor of the equation (x² -5x) ² -30 (x² -5x) -216=0?

Ans.  First of all, let x²–5x =y then equation become

y²–30y-216=0 then factorize 216 in two term for example 216=36×6

so, can be written 30y = 36y-6y

so y²-(36y-6y)-2016=0             y² -36y+6y-216=0

taking common we get

y(y-36) +6(y-36) =0       so     (y-36) (y-6) =0

then y has two values y=36 and y=6 then taking first

y=36 putting y=x²–5x

so, x²–5x-36=0 our first equation

so, factor of 36=9x4 and 9–4 =5

then x²–9x+4x-36=0

taking common x(x-9) +4(x-9) =0

(x-9) (x+4) =0 hence x=9, -4

similarly solving for y=6

Q.4. If a=x-y, b=y-z and c=z-x than what is the value of a³ + b³ +c³?

Ans:  Given a=x -y, b=y-z and c=z-x adding these equation

We get

Now, abc=x-y+y-z+z-x=0

Since we know the formula if

  a³+b³+c³- abc = (a+b+c) (a²+b²+c²-ab-bc-ca)

here a+b+c=0 so

..a³+b³+c³= abc = 3 (x-y) ( y-z) (z-x) so

a³+b³+c³=3(x-y) (y-z) (z-x)