Question

If the zeroes of a cubic polynomial are in AP and sum
and product of the zeroes are respectively 3 and -8,
then the cubic polynomial whose zeroes are the
reciprocal of the zeroes of the given cubic
polynomial can be

​

Answer 1

Answer:

If α1, α2,α3 ... αn are the roots of the equation

f(x)= a0xn +a1xn-1 +a2xn-2 +...+an-1x + an =0, then

f(x)= a0 (x-α1)(x-α2)(x-α3)... (x-αn)

Equating both the RHS terms we get,

a0xn +a1xn-1 +a2xn-2 +...+an-1x + an = a0(x-α1)(x-α2)(x-α3)... (x-αn)

Comparing coefficients of xn-1 on both sides, we get

S1 = α1 + α2+α3 +... + αn = ∑αi = -a1/ a0

or, S1= - coeff. of xn-1/coeff. of xn

Comparing coefficients of xn-2 on both sides, we get

S2 = α1 α2+ α1α3 +... = ∑αi αj = (-1)2a2/ a0

i≠ j

or, S2= (-1)2 coeff. of xn-2/coeff. of xn

Comparing coefficients of xn-3 on both sides, we get

S3 = α1 α2α3+ α2α3α4 +... = ∑αi αj αk = (-1)3a3/ a0

i≠ j≠ k

or, S3= (-1)3 coeff. of xn-3/coeff. of xn

... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ...

Sn=α1α2α3... αn =(-1)nan/ a0= (-1)n constant term/coeff. of xn

Here, Sk denotes the sum of the products of the roots taken k at a time.

For example, S3 denotes the sum of the product of roots taken 3 at a time.

PARTICULAR CASES:

Quadratic Equation:

If α and β are roots of the quadratic equation ax2 + bx + c=0, then

α + β = -b/a

α * β = c/a

Cubic Equation:

If α , β, γ are roots of a cubic equation ax3 + bx2 + cx + d=0, then

α + β + γ = -b/a

α β +β γ + γα = c/a

α βγ = -d/a

Biquadratic equation :

If α , β, γ, δ are roots of a cubic equation ax4+ bx3 + cx2 + dx +e=0, then

α + β + γ + δ = -b/a

α β +β γ + γδ +αγ + αδ+ βδ = c/a

α β γ + αγδ +αβδ + βγδ = -d/a

αβγδ = e/a

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