If $x+2$ is the only factor of the polynomial $P(x)$, then $P(-2)$ is:
A. Zero
B. $P(2)$
C. Not Zero
D. Cannot be determined
Answer (2)
Since x + 2 is the only factor of the polynomial P ( x ) , we have:
P ( x ) = k ( x + 2 ) n for some constant k and positive integer n .
Substituting x = 2 , we get P ( 2 ) = k ( 4 ) n .
Since k and 4 n are non-zero, P ( 2 ) is non-zero.
Therefore, P ( 2 ) is Not Zero .
Explanation
Understanding the Problem We are given that x + 2 is the only factor of the polynomial P ( x ) . This means that P ( x ) can be written in the form P ( x ) = k ( x + 2 ) n , where k is a constant and n is a positive integer. Our goal is to determine the value of P ( 2 ) .
Substituting x=2 To find P ( 2 ) , we substitute x = 2 into the expression for P ( x ) : P ( 2 ) = k ( 2 + 2 ) n = k ( 4 ) n = k ⋅ 4 n
Analyzing the Result Since k is a constant and 4 n is always non-zero for any positive integer n , the product k v er t 4 n is non-zero, unless k = 0 . However, if k = 0 , then P ( x ) = 0 for all x , which means P ( x ) has infinitely many factors, contradicting the statement that x + 2 is the only factor. Therefore, k must be non-zero.
Conclusion Since k is non-zero and 4 n is non-zero, their product P ( 2 ) = k v er t 4 n is also non-zero. Therefore, P ( 2 ) is not zero.
Examples
Polynomial factorization is a fundamental concept in algebra and is used in various applications, such as solving equations, curve fitting, and signal processing. For example, in control systems, the characteristic equation of a system is a polynomial, and the stability of the system depends on the roots of this polynomial. If we know the factors of the polynomial, we can easily find the roots and determine the stability of the system. Similarly, in cryptography, polynomial factorization is used in the design of public-key cryptosystems.
The polynomial P ( x ) can be expressed as P ( x ) = k ( x + 2 ) n . Evaluating P ( − 2 ) results in P ( − 2 ) = 0 , confirming that P ( − 2 ) is zero as expected since x + 2 is a factor. Therefore, the answer is A. Zero.
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