You visited us 2 times! Enjoying our articles? Unlock Full Access!

Introduction to Quadratic Equations and Polynomials

The number of...

Question

The number of polynomials $p (x)$ with integer coefficients such that the curve $y = p (x)$ passes through $(2, 2)$ and $(4, 5)$ is

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

more than 1 but finite

No worries! We‘ve got your back. Try BYJU‘S free classes today!

infinite

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A 0
Let $y = p (x) = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . . + a_{n} x^{n}$
We know,
$p (2) = 2 = a_{0} + 2 a_{1} + 2^{2} a_{2} + . . . . . + 2^{n} a_{n} p (4) = 5 = a_{0} + 4 a_{1} + 4^{2} a_{2} + . . . . . + 4^{n} a_{n}$
Now,
$p (4) - p (2) = (4 - 2) a_{1} + (4^{2} - 2^{2}) a_{2} + . . . . . . + (4^{n} - 2^{n}) a_{n}$
$3 = (4 - 2) a_{1} + (4^{2} - 2^{2}) a_{2} + . . . . . . + (4^{n} - 2^{n}) a_{n}$
From the above equation, we can see that the RHS will always remain even in nature, while the LHS is odd.
So they will never be equal.
Hence, there will be no polynomial which sastisfies the given condition.

Suggest Corrections

Similar questions

p (x)

y = p (x)

(2, 2)

(4, 5)

p (x)

p (2) = 3

p (6) = 5

P (x)

P ({sin}^{2} x) = P ({cos}^{2} x)

x ϵ [0, π / 2]

P (x)

P (x)

(2 x - 1)^{2}

P (x)

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

Q (x) = b_{n} x^{n} + b_{n - 1} x^{n - 1} + \dots + b_{1} x + b_{0}

a_{n} - b_{n}

a_{n - 1} = b_{n - 1}

a_{n} b_{0} - a_{0} b_{n} \neq 0

r

P (r) = Q (r) = 0

r

p (x)

p (x) - p^{'} (x) = x^{n}

n

p (0)

Join BYJU'S Learning Program