If the divisor $(x-a)$ is a factor of the dividend $P(x)$, then the remainder $R(a)$ will be:
A. None of these
B. Zero
C. Non-Zero
D. P(2)
Answer (1)
If ( x − a ) is a factor of the polynomial P ( x ) , then by the Remainder Theorem, the remainder when P ( x ) is divided by ( x − a ) is zero. Therefore, R ( a ) = 0 , meaning the remainder is Z ero .
Explanation
Understanding the Problem We are given that ( x − a ) is a factor of the polynomial P ( x ) . We need to find the remainder when P ( x ) is divided by ( x − a ) .
Applying the Remainder Theorem The Remainder Theorem states that if we divide a polynomial P ( x ) by ( x − a ) , the remainder is P ( a ) . Since ( x − a ) is a factor of P ( x ) , this means that P ( x ) is divisible by ( x − a ) with no remainder. In other words, ( x − a ) divides P ( x ) exactly.
Determining the Remainder Therefore, the remainder R ( a ) when P ( x ) is divided by ( x − a ) is zero. This is because if ( x − a ) is a factor of P ( x ) , then P ( a ) = 0 .
Conclusion Thus, the remainder R ( a ) will be zero.
Examples
Consider a scenario where you are designing a bridge and need to ensure that a certain support beam can handle the load applied to it. If the load can be represented by a polynomial P ( x ) and the strength of the beam is related to ( x − a ) , where a is a critical value, then knowing that ( x − a ) is a factor of P ( x ) (meaning the beam's strength is sufficient) tells you that the remainder is zero, indicating no additional stress beyond what the beam can handle. This ensures the bridge's safety and stability.
