If one of the zeros of the cubic polynomial


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If one of the zeros of the cubic polynomial x^3+ax^2+bx+c is -1, then find the product of other two zeros.

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Product of other two zeroes in the equation is –b.  

Solution:

To find the zeroes of the cubic equation x^{3}+a x^{2}+b x+c, let us say that the roots are \alpha,\beta,\gamma

The expressions of roots are a x^{3}+b x^{2}+c x+d

\begin{array}{l}{\alpha+\beta+\gamma=-\frac{b}{a}} \\ \\{\alpha \beta \gamma=\frac{c}{a}} \\ \\{\alpha \beta+\beta \gamma+\gamma \alpha=-\frac{d}{a}}\end{array}

Now one of the root is -1, putting α=-1 and a=1,b=a,c=b,d=c in the expression,

\begin{array}{l}{-1+\beta+\gamma=-a} \\ \\{-1 \beta \gamma=b} \\ \\{-1 \beta+\beta \gamma-1 \gamma=-c}\end{array}

To find the product of other zeroes we use -1 \beta \gamma=b

we get  \bold{\beta y=-b} .


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