Name: Zeros of a Polynomial
Uploaded: 2019-07-19
Duration: 13 min 14 s
Description: The zero of a polynomial is the value of variables at which the polynomial’s value is zero. To find zeroes of a polynomial, equate the polynomial to zero and...

Question

Given that one root of

2 x^{3} + 3 x^{2} - 8 x + 3 = 0

is double the other root, find the roots of the equation.

Accepted Answer

Given $2 x^{3} + 3 x^{2} - 8 x + 3 = 0$

Let $α, β, γ$ are the roots of $2 x^{3} + 3 x^{2} - 8 x + 3 = 0$ .

Here, $α + β + γ = - \frac{3}{2}$ ....(i)

$α β + β γ + γ α = - 4$ .......(ii)

$α β γ = - \frac{3}{2}$ .........................(iii)

Given $α = 2 β$

$3 β + γ = - \frac{3}{2} \Rightarrow γ = - \frac{3}{2} - 3 β$ (from equation(i)) then

$α β + γ (α + β) = - 4$ (from equation(ii))

$\Rightarrow 2 β^{2} + 3 β γ = - 4$

$\Rightarrow 2 β^{2} + 3 β (- \frac{3}{2} - 3 β) = - 4$

$\Rightarrow 2 β^{2} - \frac{3 β (3 + 6 β)}{2} = - 4$

$\Rightarrow 4 β^{2} - 9 β - 18 β^{2} = - 8$

$\Rightarrow (2 β - 1) (7 β + 8) = 0$

$\Rightarrow β = \frac{1}{2} or β = - \frac{8}{7}$

If $β = \frac{1}{2}$ , then

$α = 2 β = 2 (\frac{1}{2}) = 1$

$γ = - \frac{3}{2} - 3 β = - \frac{3}{2} - 3 (\frac{1}{2})$

$= - \frac{3}{2} - \frac{3}{2} = \frac{- 6}{2} = - 3$

$α β γ = (1) (\frac{1}{2}) (- 3) = - \frac{3}{2}$ is satisfied

Therefore, the roots are $1, \frac{1}{2}, - 3$ .

If $β = - \frac{8}{7}$ , then

$\Rightarrow$ It does not satisfy the given equation

Therefore, the roots are $\frac{1}{2}, 1, - 3$ .

Given that one root of $2 x^{3} + 3 x^{2} - 8 x + 3 = 0$ is double the other root, find the roots of the equation.

Important Questions on Theory of Equations

Learn and Prepare for any exam you want !

Given that one root of 2x3+3x2-8x+3=0 is double the other root, find the roots of the equation.

Important Questions on Theory of Equations

Learn and Prepare for any exam you want !

Given that one root of $2 x^{3} + 3 x^{2} - 8 x + 3 = 0$ is double the other root, find the roots of the equation.