Question

What must be subtracted from f x is equal to a x to the power 4 + under root ax cube + x square + ax + root a so that it is exactly divisible by X square + under root a

Answer 1

Answer:

a²

Step-by-step explanation:

What must be subtracted from f x is equal to a x to the power 4 + under root ax cube + x square + ax + root a so that it is exactly divisible by X square + under root a

f(x) = ax⁴ + √a x³ + x² + ax + √a

g(x) = x² + √a

Let say q(x) = (bx² + cx + d)

Then r is the remainder which need to be subtracted

f(x) = g(x) . q(x)  + r(x)

=>  ax⁴ + √a x³ + x² + ax + √a = (x² + √a) * (bx² + cx + d)  + r(x)

=>  ax⁴ + √a x³ + x² + ax + √a = bx⁴ + cx³ + (d + √a * b)x² + √acx + √ad  + r(x)

Comparing

b = a

c = √a

1 = d + √ab => 1 = d + √aa => d = (1 - a√a)

g(x) . q(x) = (x² + √a) * (ax² + √ax + 1 - a√a)

= ax⁴ + √ax³ + x² - a√ax² + a√ax² + ax + √a - a²

= ax⁴ + √ax³ + x²  + ax + √a - a²

r = f(x) - g(x)q(x) = ax⁴ + √a x³ + x² + ax + √a - (ax⁴ + √ax³ + x² + ax + √a - a²)

= a²

a² must be subtracted from f(x)