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Properties of Determinants

If a1xn + a2x...

Question

If $a_{1} x^{n} + a_{2} x^{n - 1} + . . . + a_{n} x$ is a zero polynomial then $a_{1} + a_{2} + . . . + a_{n}$ is .
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a_{1} x^{n} + a_{2} x^{n - 1} + . . . + a_{n} x

is a zero polynomial, then the degree of this polynomial is .

x + 1

f (x) = a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + \dots + a_{n} = 0

Q. In the polynomial

f (x) = a_{0} x^{n} + a_{1} x^{n - 1} + . . . + a_{n - 1} x + a_{n}, where a_{0}, a_{1}, a_{2}, . . ., a_{n}

are the real numbers, if

a_{0} = a_{1} = a_{2} = . . . = a_{n - 1} = 0 and a_{n} \neq 0,

a_{1}, a_{2}, a_{3}, \dots, a_{n}; (n \geq 2)

(n - 1) a_{1}^{2} - 2 n a_{2} < 0,

then atleast roots of the equation

x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + \dots + a_{n} = 0

, are imaginary.

x + 1

f (x) = a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + \dots + a_{n} = 0

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Properties of Determinants

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