Which of the following is divisible by x 2 - y 2 , ∀ x ≠ y ?

x n - y n x 2 n + 1 + y 2 n + 1 ∀ n ∈ N

x n - y n x m + y m ∀ m , n ∈ N

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Let $P (x) = 0$ be polynomial equation of degree five with integral coefficient that has at-least one integral root. If $P (2) = 13$ and $P (10) = 5 .$ Then find a value of $x$ which satisfies $p (x) = 0$ :

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Important Questions on Theory of Equation

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

f (x) = 2 x^{3} + {m x}^{2} - 13 x + n

and

2, 3

are roots of the equation

f (x) = 0

, then the values of

m

and

n

are -

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

Let

P

be a non-zero polynomial such that

P (1 + x) = P (1 - x)

for all real

x

and

P (1) = 0

. Let

m

be the largest integer such that

{(x - 1)}^{m}

divides

P (x)

for all such

P (x)

. Then,

m

equals

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

x + 2

is a factor of

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

Let

a, b, c

be non-zero real roots of the equation

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

Then

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

The number of real roots of the equation,

e^{4 x} + e^{3 x} - 4 e^{2 x} + e^{x} + 1 = 0

is:

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

The number of ordered pairs

(x, y)

of integers satisfying

x^{3} + y^{3} = 65

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Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

Let $f (x) = x^{6} - 2 x^{5} + x^{3} + x^{2} - x - 1$ and $g (x) = x^{4} - x^{3} - x^{2} - 1$ be two polynomials. Let $a, b, c$ and $d$ be the roots of $g (x) = 0$ . Then, the value of

$f (a) + f (b) + f (c) + f (d)$ is

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Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

a + b + c = 1, a b + b c + c a = 2

and

a b c = 3,

then the value of

a^{4} + b^{4} + c^{4}

is equal to:

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

Which of the following is divisible by

x^{2} - y^{2}, \forall x \neq y ?

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

The number of cubic polynomials

P (x)

satisfying

P (1) = 2, P (2) = 4, P (3) = 6, P (4) = 8

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

The value of

a

for which

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

has more than two roots, is:

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

Let

p (x)

be a quadratic polynomial such that

p (0) = 1 .

p (x)

leaves remainder

4

when divided by

x - 1

and it leaves remainder

6

when divided by

x + 1

then:

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

Let

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

be a polynomial in which

a_{i}

is a non-negative integer for each

i \in \{0, 1, 2, 3, \dots, n\}

. If

P (1) = 4

and

P (5) = 136

, what is the value of

P (3)

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

Let

r

be the remainder when

2021^{2020}

is divided by

2020^{2}

. Then

r

lies between

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

x = \sqrt{2} + \sqrt{3} + \sqrt{6}

is a root of

x^{4} + a x^{3} + b x^{2} + c x + d = 0

, where

a, b, c, d

are integers, what is the value of

|a + b + c + d|

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

Let

f (x)

be a polynomial of degree

3

such that

f (k) = - \frac{2}{k}

for

k = 2, 3, 4, 5 .

Then the value of

52 - 10 f (10)

is equal to _____ .

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

Let

r (x)

be the remainder when the polynomial

x^{135} + x^{125} - x^{115} + x^{5} + 1

is divided by

x^{3} - x .

Then

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

The number of integers

x

satisfying

- 3 x^{4} + det [\begin{matrix} 1 & x & x^{2} \\ 1 & x^{2} & x^{4} \\ 1 & x^{3} & x^{6} \end{matrix}] = 0

is equal to

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

If the sum of the coefficients of all even powers of

x

in the product

(1 + x + x^{2} + \dots + x^{2 n}) (1 - x + x^{2} - x^{3} + \dots + x^{2 n})

61,

then

n

is equal to

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Expressions, Equations and Identities

x + \frac{1}{x} = 5

, then

(x^{3} + \frac{1}{x^{3}}) - 5 (x^{2} + \frac{1}{x^{2}}) + (x + \frac{1}{x})

is equal to

Let Px=0 be polynomial equation of degree five with integral coefficient that has at-least one integral root. If P2=13 and P10=5. Then find a value of x which satisfies px=0:

Important Questions on Theory of Equation

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Let $P (x) = 0$ be polynomial equation of degree five with integral coefficient that has at-least one integral root. If $P (2) = 13$ and $P (10) = 5 .$ Then find a value of $x$ which satisfies $p (x) = 0$ :