Get 2 FREE Instant-Explanations on Filo with code FILOAPP

Prove that polynomial x^{4p} + x^{4q+1} +x^{4r+2} +x^{4s+3} ca

Question

Prove that polynomial $x^{4 p} + x^{4 q + 1} + x^{4 r + 2} + x^{4 s + 3}$ can be divided by $x^{3} + x^{2} + x + 1$ , where $p, q, r, s \in N$ .

Views: 527 students

Found 4 tutors discussing this question

Amelia Discussed

Prove that polynomial

x^{4 p} + x^{4 q + 1} + x^{4 r + 2} + x^{4 s + 3}

can be divided by

x^{3} + x^{2} + x + 1

, where

p, q, r, s \in N

14 mins ago

Discuss this question LIVE

14 mins ago

Text solutionVerified

Let

f (x) = x^{4 p} + x^{4 q + 1} + x^{4 r + 2} + x^{4 s + 3}

.
Now,

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

f (i) = i^{4 p} + i^{4 q + 1} + i^{4 r + 2} + i^{4 s + 3} = 1 + i^{1} + i^{2} + i^{3} = 1 + i - i = 0

f (- i) = (- i)^{4 p} + (- i)^{4 q + 1} + (- i)^{4 r + 2} + (- i)^{4 s + 3} = 1 + (- i)^{1} + (- i)^{2} + (- i)^{3} = 1 - i - 1 + i = 0

f (- 1) = (- 1)^{4 p} + (- 1)^{4 q + 1} + (- 1)^{4 r + 2} + (- 1)^{4 s + 3} = 0

Thus, by division theorem

f (x)

is divided by

x^{3} + x^{2} + x + 1

Was this solution helpful?

150

Report

One destination to cover all your homework and assignment needs

Learn Practice Revision Succeed

Instant 1:1 help, 24x7

60, 000+ Expert tutors

Textbook solutions

Big idea maths, McGraw-Hill Education etc

Essay review

Get expert feedback on your essay

Schedule classes

High dosage tutoring from Dedicated 3 experts

Trusted by 4 million+ students

Stuck on the question or explanation?

Connect with our maths tutors online and get step by step solution of this question.

231 students are taking LIVE classes

Question Text	Prove that polynomial $x^{4 p} + x^{4 q + 1} + x^{4 r + 2} + x^{4 s + 3}$ can be divided by $x^{3} + x^{2} + x + 1$ , where $p, q, r, s \in N$ .
Answer Type	Text solution:1
Upvotes	150