Question

what is remainder theorem ? explain how to use remainder thereom and factor theorem for linear polynomial .

Answer 1

The remainder theorem states that the remainder when a polynomial is divided by a linear factor is equal to the polynomial evaluated at the factor. The factor theorem states that a polynomial has a linear factor if and only if evaluating the polynomial at that value gives zero.

Step-by-step explanation:

The remainder theorem is a concept in algebra that allows us to determine the remainder when a polynomial is divided by a linear factor. It states that if we divide a polynomial f(x) by a linear polynomial x - a, the remainder will be equal to f(a).

To use the remainder theorem, we substitute the value of 'a' into the polynomial f(x) and evaluate it to find the remainder. This can be useful in finding the value of a polynomial at a specific point or in solving polynomial problems.

The factor theorem is closely related to the remainder theorem. It states that a polynomial f(x) has a linear factor x - a if and only if f(a) = 0. In other words, if a polynomial evaluates to zero when a specific value is substituted, then that value is a linear factor of the polynomial.

Answer 2

Basically the Remainder theorem states that the remainder of dividing a polynomial P(x) by (x - a) is given by P(a).

So for example if we divide x^ 2 - 2x + 7 by x - 2 the remainder will be
2^2 - 2(2) + 7 = 7..

If the remainder is 0 then the divisor will be a factor of the polynomial. This is the Factor Theorem and can be used to test if a given polynomial has a factor x-a.