Question

If $p(x)$ is a polynomial of degree $\geq 1$ and $a$ is any real number, then the remainder when $p(x)$ is divided by $x-a$ is

• A. $p(a)$
• B. $p(-a)$
• C. $-p(a)$
• D. $-p(-a)$

Hint:

Remember this shortcut: for divisor $x-a$, put $x=a$ in the polynomial. For divisor $x+a$, put $x=-a$ in the polynomial.

Answer 1

The Correct Option is A

Step 1: Recall the Remainder Theorem.}
The Remainder Theorem states that if a polynomial $p(x)$ is divided by a linear factor of the form $x-a$, then the remainder is obtained by substituting $x=a$ into the polynomial. Thus, the remainder is: \[ p(a) \] This is a standard result in polynomial theory.

Step 2: Apply the theorem to the given expression.}
Here, the divisor is: \[ x-a \] So, according to the Remainder Theorem, the remainder when $p(x)$ is divided by $x-a$ is: \[ p(a) \] Hence, the required remainder is $p(a)$.

Step 3: Compare with the given options.}

• (A) $p(a)$: Correct. This follows directly from the Remainder Theorem.
• (B) $p(-a)$: Incorrect. This would be the remainder if the divisor were $x+a$.
• (C) $-p(a)$: Incorrect. The remainder is not negative of $p(a)$.
• (D) $-p(-a)$: Incorrect. This does not match the theorem.

So, the correct option is (A) $p(a)$.

Step 4: Conclusion.}
Therefore, whenever a polynomial $p(x)$ is divided by $x-a$, the remainder is always equal to: \[ p(a) \] Hence, the correct answer is $p(a)$.
Final Answer:} $p(a)$.