If - and - are two real roots of the equation x3 - px2 - qx - r - 0 satisfying the relation - - 1 - 0- then find the value of r2 - pr - q-

Solution: For nbsp; x^3 nbsp;+ px^2 nbsp;+ qx + r = 0 Sum of the roots α + β + γ = p nbsp; nbsp;. . . (1) αβ + βγ + nbsp;γα = q nbsp; . . . (2 ...

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Byju's Answer

Standard XII

Mathematics

Cubic Polynomial

If α and β ar...

Question

If $α$ and $β$ are two real roots of the equation $x^{3} - p x^{2} + q x + r = 0$ satisfying the relation $α β + 1 = 0$ , then find the value of $r^{2} + p r + q$ .

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Solution

Solution: For $x^{3} + p x^{2} + q x + r = 0$

Sum of the roots $α + β + γ = - p$ . . . $(1)$

$α β + β γ + γ α = q$ . . . $(2)$

$α β γ = - r$ . . . $(3)$ (Given $α β = - 1$ )

From equation $3$ ,
$(- 1) γ = - r$

$\Rightarrow γ = r$

Substituting $γ$ in equation $2$

$- 1 + (α + β) r = q$

$\Rightarrow α + β = \frac{q + 1}{r}$ . . . $(4)$

Substitute $α + β + γ = - p$

$\Rightarrow \frac{q + 1}{r} + r = - p$

$\Rightarrow q + 1 + r^{2} = - p r$

$\Rightarrow r^{2} + p r + q + 1 = 0$

$\Rightarrow r^{2} + p r + q = - 1$

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If αand β are two real roots of the equation x3−px2+qx+r=0 satisfying the relation αβ+1=0, then find the value of r2+pr+q. __

If $α$ and $β$ are two real roots of the equation $x^{3} - p x^{2} + q x + r = 0$ satisfying the relation $α β + 1 = 0$ , then find the value of $r^{2} + p r + q$ .

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