What is the importance of the remainder theorem and factor theorem?

The remainder theorem and factor theorem are very handy tools. They tell us that we can find factors of a polynomial without using long division, synthetic division, or other traditional methods of factoring.

What is the formula of remainder theorem and factor theorem?

When a polynomial is divided by x−c, the remainder is either 0 or has degree less than the degree of x−c. Since x−c is degree 1, the degree of the remainder must be 0, which means the remainder is a constant. Hence, in either case, p(x)=(x−c)q(x)+r, where r, the remainder, is a real number, possibly 0.

What is meant by remainder theorem?

The remainder theorem definition states that when a polynomial f(x) is divided by the factor (x -a) when the factor is not necessarily an element of the polynomial, then you will find a smaller polynomial along with a remainder.

Why is the factor theorem useful?

We can use the Factor Theorem to completely factor a polynomial into the product of n factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.

What is meant by factor theorem?

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem. The factor theorem states that a polynomial has a factor if and only if (i.e. is a root).

What is factor theorem example?

Answer: An example of factor theorem can be the factorization of 6×2 + 17x + 5 by splitting the middle term. In this example, one can find two numbers, ‘p’ and ‘q’ in a way such that, p + q = 17 and pq = 6 x 5 = 30. After that one can get the factors. For example, x + 2 is a factor belonging to the polynomial x2 – 4.

What is the use of factor theorem?

Factor theorem is usually used to factor and find the roots of polynomials. A root or zero is where the polynomial is equal to zero. Therefore, the theorem simply states that when f(k) = 0, then (x – k) is a factor of f(x).

What is the formula for the remainder theorem?

Remainder Theorem. Consider the polynomial g(x) of any degree greater than or equal to one and any real number c. If g(x) is divided by a linear polynomial (x-c), then the remainder is equal to g(c). That is, g (x) = (x – c) q(x) + g(c).

What is the definition of remainder theorem?

Definition of remainder theorem. : a theorem in algebra: if f(x) is a polynomial in x then the remainder on dividing f(x) by x − a is f(a)

What do we use the polynomial remainder theorem for?

The Remainder Theorem is useful for evaluating polynomials at a given value of x , though it might not seem so, at least at first blush. This is because the tool is presented as a theorem with a proof, and you probably don’t feel ready for proofs at this stage in your studies.

What is synthetic division and remainder theorem?

Remainder Theorem. To find the remainder of a polynomial divided by some linear factor, we usually use the method of Polynomial Long Division or Synthetic Division. However, the concept of the Remainder Theorem provides us with a straightforward way to calculate the remainder without going into the hassle.