Question 28

The polynomial $$P(x)=x^3+ax^2+bx+c$$ has the property that the mean of its roots, the product of its roots, and the sum of its coefficients are all equal. If the $$y$$ intercept of the graph $$y=P(x)$$ is $$2$$, then $$b$$ is

Correct Answer: -11

Solution

If the roots are $$\alpha,\beta,\gamma$$, their mean is $$-\frac{a}{3}$$ and their product is $$-c$$. The sum of the coefficients is $$1+a+b+c$$, and the $$y$$ intercept gives $$c=2$$. Thus $$-\frac{a}{3}=-2=1+a+b+2$$, which gives $$a=6$$ and $$b=-11$$.

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The polynomial $$P(x)=x^3+ax^2+bx+c$$ has the property that the mean of its roots, the product of its roots, and the sum of its coefficients are all equal. If the $$y$$ intercept of the graph $$y=P(x)$$ is $$2$$, then $$b$$ is

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