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[tex]x-a[/tex] is the factor of a polynomial [tex]P(x)[/tex] if [tex]P(a)[/tex] is equal to
A) 2
B) 3
C) 1
D) 0
Answer (2)
The problem states that x − a is a factor of polynomial P ( x ) .
Apply the Factor Theorem, which states that if x − a is a factor of P ( x ) , then P ( a ) = 0 .
Therefore, P ( a ) must be equal to 0.
The final answer is 0 .
Explanation
Understanding the Problem We are given that x − a is a factor of the polynomial P ( x ) . We need to determine the value of P ( a ) . This problem involves the Factor Theorem.
Applying the Factor Theorem The Factor Theorem states that for a polynomial P ( x ) , x − a is a factor of P ( x ) if and only if P ( a ) = 0 .
Determining the Value of P(a) Since x − a is a factor of P ( x ) , according to the Factor Theorem, P ( a ) must be equal to 0.
Final Answer Therefore, the value of P ( a ) is 0.
Examples
The Factor Theorem is used in polynomial factorization and root finding. For example, if you know that x − 2 is a factor of a polynomial P ( x ) , then you know that P ( 2 ) = 0 . This can help you find the roots of the polynomial and simplify it. In engineering, this can be used to analyze systems where the behavior is modeled by polynomials, such as control systems or signal processing. Knowing the factors helps in understanding the system's stability and response.
According to the Factor Theorem, if x − a is a factor of the polynomial P ( x ) , then P ( a ) must equal 0 . Therefore, the answer is 0 . The chosen option is (D) 0.
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