Question

when polynomial p(x) is divided by 2x+1 is it possible to have x-1 as a remainder ? justify your answer​

Answer 1

SOLUTION

TO JUSTIFY

When polynomial p(x) is divided by 2x+1 is it possible to have x - 1 as a remainder

CONCEPT TO BE IMPLEMENTED

DIVISION ALGORITHM

Let f(x) & g(x) be two polynomials of degree n and m respectively and $\sf{n \geqslant m}$ . Then there exists two uniquely determined polynomials q(x) & r(x) satisfying

f(x) = g(x) q(x) + r(x)

Where the degree of q(x) is n - m and r(x) is either a zero polynomial or the degree of r(x) is less than m

EVALUATION

Here it is given that the polynomial p(x) is divided by 2x + 1

∴ g(x) = 2x + 1

∴ degree of g(x) = 1

If possible let the Remainder = r(x) = x - 1

Now degree of r(x) = 1

So that, degree of r(x) = degree of g(x)

Hence according to Division algorithm we arrives at a contradiction as it must be always degree of r(x) < degree of g(x)

Hence When polynomial p(x) is divided by 2x+1 it is not possible to have x - 1 as a remainder

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. The polynomial f(x) = x⁴ - 2x³ - px² + q is divisible by (x-2)². Find the values p and q. Hence, solve the equation.

https://brainly.in/question/27895807

2. Find the degree of 2020?

https://brainly.in/question/25939171