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

Assume that α , β , γ are the roots of 2 x 3 + 5 x 2 + 5 x + 2 = 0 . For h ∈ ℝ , , if α + h , β + h , γ + h are roots of a h x 3 + b h x 2 + c h x + d h = 0 , then

d ( h ) vanishes for three distinct real values of h

HARD

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Monic quadratic polynomials $P (x)$ and $Q (x)$ have the property that $P (Q (x))$ has zeroes at $x = - 23, - 21, - 17$ and $- 15$ and $Q (P (x))$ has zeroes at $x = - 59, - 57, - 51,$ and $- 49 .$ What is the last two digits of the sum of the minimum values of $P (x)$ and $Q (x) ?$

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Important Questions on Polynomials

MEDIUM

Mathematics>Algebra>Polynomials>Vieta's Relations

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 =

EASY

Mathematics>Algebra>Polynomials>Vieta's Relations

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>Polynomials>Vieta's Relations

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>Polynomials>Vieta's Relations

Which of the following is divisible by

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

MEDIUM

Mathematics>Algebra>Polynomials>Vieta's Relations

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>Polynomials>Vieta's Relations

Assume that

α, β, γ

are the roots of

2 x^{3} + 5 x^{2} + 5 x + 2 = 0 .

For

h \in ℝ,

, if

α + h, β + h, γ + h

are roots of

a (h) x^{3} + b (h) x^{2} + c (h) x + d (h) = 0,

then

MEDIUM

Mathematics>Algebra>Polynomials>Vieta's Relations

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>Polynomials>Vieta's Relations

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

HARD

Mathematics>Algebra>Polynomials>Vieta's Relations

The figure shows a portion of the graph $y = 2 x - 4 x^{3} .$ The line $y = c$ is such that the areas of the regions marked $I$ and $II$ are equal. If $a, b$ are the $x$ -coordinates of $A, B$ respectively, then $a + b$ equals-

Question Image

HARD

Mathematics>Algebra>Polynomials>Vieta's Relations

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-

MEDIUM

Mathematics>Algebra>Polynomials>Vieta's Relations

If the sum of any two roots of the equation

x^{3} + p x^{2} + q x + r = 0

is zero, then

EASY

Mathematics>Algebra>Polynomials>Vieta's Relations

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

EASY

Mathematics>Algebra>Polynomials>Vieta's Relations

a, b

and

c

are the roots of

x^{3} + 4 x + 1 = 0,

then

\frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{c + a} =

MEDIUM

Mathematics>Algebra>Polynomials>Vieta's Relations

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>Polynomials>Vieta's Relations

The number of cubic polynomials

P (x)

satisfying

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

HARD

Mathematics>Algebra>Polynomials>Vieta's Relations

Let

p_{1} (x) = x^{3} - 2020 x^{2} + b_{1} x + c_{1}

and

p_{2} (x) = x^{3} - 2021 x^{2} + b_{2} x + c_{2}

be polynomials having two common roots

α

and

β

. Suppose there exist polynomials

q_{1} (x)

and

q_{2} (x)

such that

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

. Then the correct identity is

EASY

Mathematics>Algebra>Polynomials>Vieta's Relations

x + 2

is a factor of

HARD

Mathematics>Algebra>Polynomials>Vieta's Relations

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

MEDIUM

Mathematics>Algebra>Polynomials>Vieta's Relations

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>Polynomials>Vieta's Relations

2 + 3 i

is one of the roots of the equation

2 x^{3} - 9 x^{2} + k x - 13 = 0, k \in R,

then the real root of this equation (where

i^{2} = - 1

) :

Monic quadratic polynomials P(x) and Q(x) have the property that P(Q(x)) has zeroes at x=-23,-21,-17 and -15 and Q(P(x)) has zeroes at x=-59,-57,-51, and -49. What is the last two digits of the sum of the minimum values of P(x) and Q(x)?

Important Questions on Polynomials

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Monic quadratic polynomials $P (x)$ and $Q (x)$ have the property that $P (Q (x))$ has zeroes at $x = - 23, - 21, - 17$ and $- 15$ and $Q (P (x))$ has zeroes at $x = - 59, - 57, - 51,$ and $- 49 .$ What is the last two digits of the sum of the minimum values of $P (x)$ and $Q (x) ?$