Question

Can $\left( x-1 \right)$ be the remainder on division of a polynomial p(x) by \[2x+3\]?

Answer 1

Hint: In this question, we are asked to find out whether there is any polynomial p(x) such that when it is divided by the polynomial \[2x+3\], the remainder is equal to \[x-1\]. To solve this question, we will need to use the properties of the remainder in the division of polynomials. Thereafter, we can check whether there is any valid polynomial p(x) whose division with the divisor polynomial $2x+3$ yields a remainder $x-1$.

Complete step by step answer:
In this question, we have to check if there is a polynomial p(x) such that when it is divided by the polynomial \[2x+3\], the remainder is equal to \[x-1\]. Therefore, here the divisor is the polynomial $2x+3$and the remainder is the polynomial $x-1$.
The degree of a polynomial is defined to be the highest power x that appears in it when it is expressed in terms of monomials i.e. sum of terms with different powers of x with non-zero coefficients…………… (1.1)
Now, the theorem for polynomial division states that
When a polynomial p(x) is divided by a divisor g(x), it can be written in the form
$p(x)=q(x)g(x)+r(x)$
Where the polynomial q(x) is called the quotient and the polynomial r(x) is known as the remainder and either r(x)0 or the degree of r(x) should be less than that of g(x)……………. (1.2)

In this case the divisor g(x) is given to be $2x+3$. By using equation (1.1), we find that the degree of g(x) is 1.
The remainder is given to be $r(x)=x-1$ whose degree is 1 according to equation (1.1).
Thus, we find that the degree of g(x)= degree of r(x)= 1……………………. (1.3)
However, from equation (1.2), the degree of r(x) should be lower than that of g(x) which contradicts equation (1.3). Thus, no such polynomial is possible which on division by $2x+3$ gives remainder $x-1$.

Note: We should correctly identify the divisor and quotient in the given question and then apply equation (1.2). The divisor should be the one by which the polynomial is divided and remainder is the term remaining added to the product of divisor and quotient to obtain the polynomial.