Question

How do you know if a polynomial is not factorable?

Answer 1

Hint: In this question, we need to know if a polynomial is factorable or non-factorable. For this, we can use the polynomial remainder theorem. If the remainder is zero then the polynomial is factorable and if the remainder exists, then the polynomial is non-factorable according to the factor theorem.

Complete step-by-step answer:
For us, it is important for us to know whether the given polynomial is factorable or no-factorable before solving it. This identification is very useful when it comes to higher degrees like $ 3,4,5, \ldots $ and so on. Since, we can’t waste time in solving the polynomials with the higher degree, as it will take longer calculations and more time.
Hence, to find the given polynomial is factorable or non-factorable, we can use the polynomial remainder theorem. It states that the remainder of the division of a polynomial $ f\left( x \right) $ by a linear polynomial $ x - r $ is equal to $ f\left( r \right) $ . $ x - r $ is a divisor of $ f\left( x \right) $ if and only if $ f\left( r \right) = 0 $ , which is a property known as factor theorem. If $ f\left( r \right) = 0 $ , then $ r $ is a root of the polynomial $ f\left( x \right) $ .
Therefore, if the remainder of the polynomial is zero then we can say the given polynomial is factorable, if the remainder is not zero, then we can say it is non-factorable.

Note: 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 polynomial remainder theorem is also known as little Bezout’s theorem, which is an application of Euclidean division of polynomials. Euclidean division of polynomials, which is used in Euclid’s algorithm for computing GCDs, is very similar to Euclidean division of integers.