Question

when a poly nomial x^6+x^4-x^2+1 is divided by x-2, then find the reminder

Answer 1

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What's The Remainder When Divided by (x-1)(x-2)

polynomials

If a polynomial is divided by (x−1)(x−1) then remainder is 5 and if divided by (x−2)(x−2) the remainder is 7. What will be the remainder is the polynomial is divided by (x−1)(x−2)(x−1)(x−2) ?

As the degree is unknown so we can't write the polynomial with arbitrary coefficients. So we have to assume the polynomial as f(x)f(x). Now we can write ...

f(x)=(x−1)g(x)+5f(x)=(x−1)g(x)+5

f(x)=(x−2)h(x)+7f(x)=(x−2)h(x)+7

Where g(x)g(x) & h(x)h(x) are some polynomial of x. then I subtract these two equations, but can't go further. Am I going correct? Should I need to use differentiation?

Answer 2

given polynomial = x^6+x^4-x^2+1
g(x) = x-2 = 0
x = 2
p(2) = (2)^6+(2)^4-(2)^2+1
= 64+16-4+1
= 77