**If a polynomials f(x) is divided by (x-a) i.e. a polynomial of degree 1 then the remainder is f(a) .**

We know that

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

where degree r(x) < degree g(x)

choose g(x) = (x-a)

there fore f(x) = (x-a) q(x) + r(x)

where degree r(x) < degree g(x) , i.e. <1 , or degree r(x) =0 or say r(x) = r .

therefore , f(x) = (x-a) q(x) + r

therefore , f(a) = (a-a) q(a) + r

or f(a) = r = remainder when the polynomial f(x) is divided by x-a .

therefore , f(x) = (x-a) q(x) +f(a)

Here is an example of Reminder Theorem

Show that the polynomial remainder theorem holds for an arbitrary second degree polynomial by using algebraic manipulation:

*x*−

*r*) gives

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