If $1, 1, \alpha$ are the roots of $x^3 - 6x^2 + 9x - 4 = 0$, then find $\alpha$.

If 1, 1, α are the roots of x³ - 6x² + 9x - 4 = 0, – Class 12 Mathematics

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1, 1, \alpha

Yanam BieapBIEAP Intermediate Board 2018Subjective· 2mImportance★★★★★

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Use the relation between roots and coefficients (sum of roots) of the cubic to find $\alpha$ .

For a cubic $x^3+bx^2+cx+d=0$ , the sum of the roots equals $-b$ . Here $x^3-6x^2+9x-4=0$ , so $b=-6$ , and:

$\text{sum of roots} = -(-6) = 6$

The roots are $1,1,\alpha$ , so:

$1+1+\alpha = 6 \;\Rightarrow\; \alpha = 4$ …