Use the remainder theorem to find the remainder when [tex]$f(x)$[/tex] is divided by [tex]$x-2$[/tex]. Then use the factor theorem to determine whether [tex]$x-2$[/tex] is a factor of [tex]$f(x)$[/tex].
[tex]$f(x)=2 x^3-2 x^2-5 x+9$[/tex]
Answer (1)
Use the Remainder Theorem to find the remainder when f ( x ) is divided by x − 2 by evaluating f ( 2 ) .
Calculate f ( 2 ) = 2 ( 2 ) 3 − 2 ( 2 ) 2 − 5 ( 2 ) + 9 = 7 .
Apply the Factor Theorem: since f ( 2 ) = 7 e q 0 , x − 2 is not a factor of f ( x ) .
The remainder is 7 and x − 2 is not a factor of f ( x ) , so the remainder is 7 .
Explanation
Understanding the Problem We are given the polynomial f ( x ) = 2 x 3 − 2 x 2 − 5 x + 9 . We want to find the remainder when f ( x ) is divided by x − 2 using the Remainder Theorem, and then determine if x − 2 is a factor of f ( x ) using the Factor Theorem.
Applying the Remainder Theorem The Remainder Theorem states that if we divide a polynomial f ( x ) by x − c , then the remainder is f ( c ) . In this case, we want to divide by x − 2 , so c = 2 . We need to find f ( 2 ) .
Calculating f(2) We evaluate f ( 2 ) : f ( 2 ) = 2 ( 2 ) 3 − 2 ( 2 ) 2 − 5 ( 2 ) + 9 f ( 2 ) = 2 ( 8 ) − 2 ( 4 ) − 10 + 9 f ( 2 ) = 16 − 8 − 10 + 9 f ( 2 ) = 7
Applying the Factor Theorem The Factor Theorem states that x − c is a factor of f ( x ) if and only if f ( c ) = 0 . Since f ( 2 ) = 7 e q 0 , x − 2 is not a factor of f ( x ) .
Conclusion Therefore, the remainder when f ( x ) is divided by x − 2 is 7, and x − 2 is not a factor of f ( x ) .
Examples
Polynomials and the Remainder Theorem are useful in various applications, such as determining if a certain value is a root of a polynomial, which can be used in engineering to analyze the stability of systems. For example, if the characteristic equation of a system is given by a polynomial, we can use the Remainder Theorem to check if a particular value is a root, which indicates the system's stability. In computer graphics, polynomials are used to model curves and surfaces, and the Remainder Theorem can help in simplifying these models or finding specific points on them. Understanding these concepts allows engineers and designers to create more efficient and stable systems and models.
