Let P ( x ) be a polynomial such that x · P ( x - 1 ) = ( x - 4 ) P ( x ) ∀ x ∈ R Find all such P ( x ) .

Given: x · P x - 1 = x - 4 P x ∀ x ∈ R Put x = 0 , 0 = - 4 P ( 0 ) ⇒ P ( 0 ) = 0 Put x = 1 , then 1 · P 0 = - 3 P 1 ⇒ P 1 = 0 Put x = 2 , 2 · P ( 1 ) = - 2 P ( 2 ) ⇒ P ( 2 ) = 0 Put x = 3 , 3 · P ( 2 ) = - P ( 3 ) ⇒ P ( 3 ) = 0 Let us assume P ( x ) = x ( x - 1 ) ( x - 2 ) ( x - 3 ) Q ( x ) , where Q ( x ) is some polynomial Now using given relation we have x ( x - 1 ) ( x - 2 ) ( x - 3 ) ( x - 4 ) Q ( x - 1 ) = x ( x - 1 ) ( x - 2 ) ( x - 3 ) ( x - 4 ) Q ( x ) ⇒ Q ( x - 1 ) = Q ( x ) ∀ x ∈ R - { 0 , 1 , 2 , 3 , 4 } ⇒ Q ( x - 1 ) = Q ( x ) ∀ x ∈ R (From identity theorem) ⇒ Q ( x ) is periodic ⇒ Q ( x ) = c ⇒ P ( x ) = c x ( x - 1 ) ( x - 2 ) ( x - 3 )

Let P ( x ) be a polynomial such that x · P ( x - 1 ) = ( x - 4 ) P ( x ) ∀ x ∈ R Find all such P ( x ) .

Given: x · P x - 1 = x - 4 P x ∀ x ∈ R Put x = 0 , 0 = - 4 P ( 0 ) ⇒ P ( 0 ) = 0 Put x = 1 , then 1 · P 0 = - 3 P 1 ⇒ P 1 = 0 Put x = 2 , 2 · P ( 1 ) = - 2 P ( 2 ) ⇒ P ( 2 ) = 0 Put x = 3 , 3 · P ( 2 ) = - P ( 3 ) ⇒ P ( 3 ) = 0 Let us assume P ( x ) = x ( x - 1 ) ( x - 2 ) ( x - 3 ) Q ( x ) , where Q ( x ) is some polynomial Now using given relation we have x ( x - 1 ) ( x - 2 ) ( x - 3 ) ( x - 4 ) Q ( x - 1 ) = x ( x - 1 ) ( x - 2 ) ( x - 3 ) ( x - 4 ) Q ( x ) ⇒ Q ( x - 1 ) = Q ( x ) ∀ x ∈ R - { 0 , 1 , 2 , 3 , 4 } ⇒ Q ( x - 1 ) = Q ( x ) ∀ x ∈ R (From identity theorem) ⇒ Q ( x ) is periodic ⇒ Q ( x ) = c ⇒ P ( x ) = c x ( x - 1 ) ( x - 2 ) ( x - 3 )

Let a ∈ R and let f : R → R be given by f x = x 5 - 5 x + a , then

f x has three real roots if - 4 < a < 4

f x has three real roots if a > 4

f x has three real roots if a < - 4

MEDIUM

Earn 100

Let $P (x)$ be a polynomial such that $x \cdot P (x - 1) = (x - 4) P (x) \forall x \in R$ Find all such $P (x)$ .

Important Questions on Polynomials

MEDIUM

Mathematics>Algebra>Polynomials>Polynomial Equations

f (x)

is a polynomial of degree

n

with rational coefficients and

1 + 2 i, 2 - \sqrt{3}

and

5

are three roots of

f (x) = 0,

then the least value of

n

MEDIUM

Mathematics>Algebra>Polynomials>Polynomial Equations

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>Polynomials>Polynomial Equations

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>Polynomials>Polynomial Equations

x - 3

and

x + 3

are the factors of

4 x^{3} + a x^{2} + b x

, then the values of

a

and

b

are respectively -

HARD

Mathematics>Algebra>Polynomials>Polynomial Equations

Let

a \in R

and let

f : R \to R

be given by

f (x) = x^{5} - 5 x + a,

then

MEDIUM

Mathematics>Algebra>Polynomials>Polynomial Equations

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>Polynomials>Polynomial Equations

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

HARD

Mathematics>Algebra>Polynomials>Polynomial Equations

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)

HARD

Mathematics>Algebra>Polynomials>Polynomial Equations

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) ?

EASY

Mathematics>Algebra>Polynomials>Polynomial Equations

Let

f (x) = x^{13} + x^{11} + x^{9} + x^{7} + x^{5} + x^{3} + x + 12

. Then

MEDIUM

Mathematics>Algebra>Polynomials>Polynomial Equations

A polynomial of odd degree with real coefficients must have -

MEDIUM

Mathematics>Algebra>Polynomials>Polynomial Equations

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>Polynomials>Polynomial Equations

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>Polynomials>Polynomial Equations

The equation

x^{{(\log_{3} x)}^{2} - \frac{9}{2} \log_{3} x + 5} = 3 \sqrt{3}

has

HARD

Mathematics>Algebra>Polynomials>Polynomial Equations

The equation

166 \times 56 = 8590

is valid in some base

b \geq 10

(that is,

1, 6, 5, 8, 9, 0

are digits in base

b

in the above equation). Find the sum of all possible values of

b \geq 10

satisfying the equation.

HARD

Mathematics>Algebra>Polynomials>Polynomial Equations

Find the number of ordered triples

(x, y, z)

of real numbers that satisfy the system of equation

x + y + z = 7; x^{2} + y^{2} + z^{2} = 27; x y z = 5

HARD

Mathematics>Algebra>Polynomials>Polynomial Equations

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|

MEDIUM

Mathematics>Algebra>Polynomials>Polynomial Equations

The number of cubic polynomials

P (x)

satisfying

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

MEDIUM

Mathematics>Algebra>Polynomials>Polynomial Equations

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

HARD

Mathematics>Algebra>Polynomials>Polynomial Equations

The polynomial equation

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

has

Let P(x) be a polynomial such that x·P(x-1)=(x-4)P(x)∀x∈R Find all such P(x).

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Let $P (x)$ be a polynomial such that $x \cdot P (x - 1) = (x - 4) P (x) \forall x \in R$ Find all such $P (x)$ .