Question

Given that the equation 12x³-4x²-5x+2 has two zeroes equal. find all the zeroes.

Answer 1

a= 12 , b= -4 ,c =-5,d= 2
further ,
you use this formulas,
a+b+y =-b/a
ab+ by +ya=c/a
a ×b ×y =-d/a

Answer 2

Let's assume that 12x^3 - 4x² -5x + 2 has zeroes

£, £ & β

By relatioship of zeroes with coefficients

£ + £ +β = -b/a = 4/12 = 1/3

=> β = 1/3 - 2£ = (1–6£) /3 ……….(1)

The 3 factors of the given polynomial =

(x - £)(x - £) (x - β)

= ( x-£)(x-£) { x- (1–6£)/3}

= ( x² - 2£x + £²)( 3x -1+6£)/3

={ 3x^3 - x² +(-9£² + 2£)x + 6£^3 - £² }/3

= 12x^3 -4x² +4(-9£² +2£) x + 24 £^3 -4£²

Now, by comparing the coefficients of the above polynomial, with the coefficients of the given polynomial..

we get, 4(-9£² + 2£) = -5

=> 36£² -8£ -5 = 0

=> (2£ -1) (18£ + 5) = 0

= > £ = 1/2 or -5/18

If £ = 1/2,

2£ +β = 1/3

=> β = -2/3

So the zeroes of the given polynomial are

1/2, 1/2 & -2/3 ………..ANS

Other value of £ , is ruled out..

And here (x- 1/2)(x-1/2) ( x +2/3) = 12x^3 - 4x² - 5x +2