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For what values of $m \in N,$ the following divisibility $\frac{x^{m} + y^{m}}{x + y}$ holds?

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Important Questions on Theory of Equations

MEDIUM

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

The number of real roots of the equation

{(x + \frac{1}{x})}^{3} + x + \frac{1}{x} = 0

HARD

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

Let

S = \{x \in R : x \geq 0 & 2 |\sqrt{x} - 3| + \sqrt{x} (\sqrt{x} - 6) + 6 = 0\}

. Then

S

EASY

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

Let

S

be the set of all real roots of the equation

3^{x} (3^{x} - 1) + 2 = |3^{x} - 1| + |3^{x} - 2|

, then

HARD

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

If the bi-quadratic equation

f (x) = x^{4} + 2 x^{3} - 16 x^{2} - 22 x + 7 = 0

has

2 + \sqrt{3}

as one of its roots, then which of the following is not a root of

f (x) ?

MEDIUM

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

Let

a, b, c, d

be real numbers between

- 5

and

5

such that

|a| = \sqrt{4 - \sqrt{5 - a}}, |b| = \sqrt{4 + \sqrt{5 - b}}, |c| = \sqrt{4 - \sqrt{5 + c}}, |d| = \sqrt{4 + \sqrt{5 + d}}

, Then the product

a b c d

HARD

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

In the real number system, the equation

\sqrt{x + 3 - 4 \sqrt{x - 1}} + \sqrt{x + 8 - 6 \sqrt{x - 1}} = 1

has

MEDIUM

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

The number of real roots of the equation

5 + |2^{x} - 1| = 2^{x} (2^{x} - 2)

is :

MEDIUM

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

Let

α, β, γ

be the roots of the equation

x^{3} + p x + q = 0

and

f (x) = 3 p^{2} x^{2} + p^{2} x + 3 q

. Then

\sum α^{2} β + \sum α^{4} =

MEDIUM

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

For a real number

x

, let

[x]

denote the largest integer less than or equal to

x

, and let

\{x\} = x - [x]

. The number of solutions

x

to be equation

[x] \{x\} = 5

with

0 \leq x \leq 2015

HARD

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

The number of integers

n

for which

3 x^{3} - 25 x + n = 0

has three real roots is-

MEDIUM

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

Which of the following is not a root of

f (x) = x^{3} - 11 x^{2} + 36 x - 36 ?

HARD

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

Suppose

a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}

are integers such that

\frac{5}{7} = \frac{a_{2}}{2!} + \frac{a_{3}}{3!} + \frac{a_{4}}{4!} + \frac{a_{5}}{5!} + \frac{a_{6}}{6!} + \frac{a_{7}}{7!}

, where

0 \leq a_{j} < j

for

j = 2, 3, 4, 5, 6 7

. The sum

a_{2} + a_{3} + a_{4} + a_{5} + a_{6} + a_{7}

EASY

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

α, β, γ

are the roots of

x^{3} - 2 x^{2} + 3 x - 4 = 0

, then find

\sum α β (α + β)

MEDIUM

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

The set of all positive real values of

k,

for which the equation

x^{3} - 9 x^{2} + 24 x - k = 0

has three distinct real roots is the interval

MEDIUM

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

α, β, γ

are the roots of the equation

x^{3} + a x^{2} - b x + c = 0

, then

\sum β^{2} (γ + α) =

MEDIUM

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

The equation

x^{5} - 5 x^{3} + 5 x^{2} - 1 = 0

has three equal roots. If

α, β

are the other two root of this equation, then

α + β + α β =

MEDIUM

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

If the roots of the equation

\sqrt{\frac{x}{1 - x}} + \sqrt{\frac{1 - x}{x}} = \frac{5}{2}

are

p

and

q (p > q)

and the roots of the equation

(p + q) x^{4} - p q x^{2} + \frac{p}{q} = 0

are

α, β, γ, δ

, then

{(Σ α)}^{2} - Σ α β + α β γ δ =

HARD

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

Let

n \geq 3

be an integer. For a permutation

σ = (a_{1}, a_{2}, \dots ., a_{n})

(1, 2, \dots ., n)

we let

f_{σ} (x) = a_{n} x^{n - 1} + a_{n - 1} x^{n - 2} + \dots + a_{2} x + a_{1}

. Let

S_{σ}

be the sum of the roots of

f_{σ} (x) = 0

and let

S

denotes the sum over all permutations

σ

(1, 2, \dots \dots, n)

of the number

S_{σ} .

Then-

HARD

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

The number of integers

a

in the interval

[1, 2014]

for which the system of equations

x + y = a, \frac{x^{2}}{x - 1} + \frac{y^{2}}{y - 1} = 4

has finitely many solutions is -

HARD

Mathematics>Algebra>Theory of Equations>Algebraic Equations of Higher Degree

The polynomial equation

x^{3} - 3 a x^{2} + (27 a^{2} + 9) x + 2016 = 0

has

For what values of m∈N, the following divisibility xm+ymx+y holds?

Important Questions on Theory of Equations

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For what values of $m \in N,$ the following divisibility $\frac{x^{m} + y^{m}}{x + y}$ holds?