Question

state and prove the reminder theorem​

Answer 1

Answer:

Hi buddy

Step-by-step explanation:

Remainder Theorem Proof:

Let f(x) be any polynomial with degree greater than or equal to 1. Further suppose that when f(x) is divided by a linear polynomial p(x) = ( x -a), the quotient is q(x) and the remainder is r(x). ... Hence the remainder when f(x) is divided by the linear polynomial (x-a) is f(a).

How do you prove Remainder Theorem?

A more general theorem is: If f(x) is divided by ax + b (where a & b are constants and a is non-zero), the remainder is f(-b/a). Proof: Let Q(x)be the quotient and R the remainder.

How do you prove remainder theorem and factor theorem?

The remainder theorem states the following: If you divide a polynomial f(x) by (x - h), then the remainder is f(h). The theorem states that our remainder equals f(h). Therefore, we do not need to use long division, but just need to evaluate the polynomial when x = h to find the remainder.

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Answer 2

$\huge{\underline{\tt{Remainder \: Theorem}}}$

Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x - a, then the remainder is p(a).

$\huge{\underline{\tt{To\: prove}}}$

Let p(x) be any polynomial with degree greater than or equal to 1. Suppose that when p(x) Is divided by x - a, the quotient is q(x) and the remainder is r(x), i.e.,

p(x) = (x - a) q(x) + r(x)

Since, the degree of x - a is 1 and the degree of r(x) is less than the degree of x - a, the degree of r(x) = 0. This means r(x) is a constant, say r.

So, for every value of x, r(x) = r.

Therefore,

p(x) = (x - a) q(x) + r

And, if x = c then this equation gives us

p(c) = (c - a) q(c) + r