Question

When a function f(x) is divided by a polynomial, the remainder is f(-p/9) . The polynomial is:

a) px + q
b) px - q
c) qx + p
d) qx - p

Answer 1

The polynomial that leaves a remainder of f(-p/9) when dividing f(x) is in the form of qx + p, where q is the coefficient of x. This corresponds to option (c) from the given choices.

Step-by-step explanation:

When a function f(x) is divided by a polynomial, and the remainder is given by f(-p/9), this implies we are dealing with the Remainder Theorem. According to this theorem, if a polynomial f(x) is divided by a linear divisor of the form x - r, the remainder of the division is f(r). In this scenario, the remainder is f(-p/9), suggesting that the divisor must have the form x - (-p/9), which simplifies to x + (p/9). Multiplying the entire expression by 9 to get rid of the fraction yields 9x + p. This indicates the correct form of the polynomial is a linear expression with x having a positive coefficient and p being added, which corresponds to option (c) qx + p if we consider q to be 9.

Therefore, the answer to the student's question is option (c) qx + p.