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Geometrical Meaning of Zeros of a Polynomial

The polynomia...

Question

The polynomial $p (x) = x^{2} + a x^{2} + b x + c$ has the property that the mean of its zeroes ,the product of its zeroes and the sum of its coefficient are all equal .If the y- intercept of the graph of y=P(X) IS $2$ , Then the value of b is

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Solution

The y intercept is the constant so $c = 2$
$\begin{matrix} p (x) = x^{3} + a x^{2} + b x + c p (x) = x^{3} + a x^{2} + b x + 2 Let the zeros be α, β, and γ α + β + γ = \frac{- a}{1} s o \frac{α + β + γ}{3} = \frac{- c}{3} α β γ = \frac{- c}{1} = - 2 \end{matrix}$
$\begin{matrix} The coefficients are 1, a and b so \frac{α + β + γ}{3} = α β γ = 1 + a + b \frac{- a}{3} = - 2 = 1 + a + b a = 6 - 2 = 1 + 6 + b - 2 = 7 + b b = - 9 \end{matrix}$

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P (x) = x^{3} + a x^{2} + b x + c

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