Question

p(x) = x⁴ + x³ + x²+x+1
g(x) = x+1 polynomials(long division)​

Answer 1

Answer:

The good old division statement reads [math]{x^4} + 1 = \left( {{x^3} - {x^2} + x - 1} \right)g(x) + 2 \Rightarrow \ left( {{x^3} - {x^2} + x - 1} \right)g(x) = {x^4} - 1 ..

The good old division statement reads [math]{x^4} + 1 = \left( {{x^3} - {x^2} + x - 1} ight)g(x) ...

Let g(x)=m, so m(x^3-x^2+x-1)+2=x^4+1 Or,mx^3-mx^2+mx-m+2=x^4+1 Or, x^4-mx^3+mx^2-mx+m-1=0 ... More

Answer 2

Answer:

p(x) = x⁴+ x³-2x²+ x + 1

By Remainder Theorem :

x - 1 = 0

x = 1

p(x) = x⁴+ x³-2x²+ x + 1

p(1) = (1)⁴ + (1)³ - 2(1)² + (1) +1

p(1) = 1 + 1 - 2 + 1

p(1) = 3 - 2

p(1) = 1

Remainder = 1