Find the value of [tex]$c$[/tex] so that the polynomial [tex]$p(x)$[/tex] is divisible by [tex]$(x-3)$[/tex].
[tex]$p(x)=-x^3+c x^2-4 x+3$[/tex]
Answer (2)
Apply Remainder Theorem: If p ( x ) is divisible by ( x − 3 ) , then p ( 3 ) = 0 .
Substitute x = 3 into p ( x ) : p ( 3 ) = − ( 3 ) 3 + c ( 3 ) 2 − 4 ( 3 ) + 3 .
Set p ( 3 ) = 0 and solve for c : − 27 + 9 c − 12 + 3 = 0 .
Find the value of c : 4 .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = − x 3 + c x 2 − 4 x + 3 and we need to find the value of c such that p ( x ) is divisible by ( x − 3 ) . This means that when we divide p ( x ) by ( x − 3 ) , the remainder is 0.
Applying the Remainder Theorem According to the Remainder Theorem, if p ( x ) is divisible by ( x − 3 ) , then p ( 3 ) = 0 . So, we need to substitute x = 3 into the polynomial and set it equal to 0.
Substitution Substituting x = 3 into p ( x ) , we get: p ( 3 ) = − ( 3 ) 3 + c ( 3 ) 2 − 4 ( 3 ) + 3 p ( 3 ) = − 27 + 9 c − 12 + 3 Since p ( 3 ) = 0 , we have: − 27 + 9 c − 12 + 3 = 0
Solving for c Now, we solve for c :
− 27 + 9 c − 12 + 3 = 0 9 c − 36 = 0 9 c = 36 c = 9 36 c = 4
Conclusion Therefore, the value of c that makes the polynomial p ( x ) divisible by ( x − 3 ) is 4.
Examples
Polynomial divisibility is a concept used in various engineering applications. For instance, when designing filters in signal processing, engineers often need to ensure that a certain polynomial representing the filter's transfer function is divisible by another polynomial to achieve desired filtering characteristics. If a filter's transfer function is given by p ( x ) = − x 3 + c x 2 − 4 x + 3 , and it must have a zero at x = 3 for proper signal attenuation at a specific frequency, we solve for c to ensure p ( x ) is divisible by ( x − 3 ) . This ensures the filter meets the required specifications.
To make the polynomial p ( x ) = − x 3 + c x 2 − 4 x + 3 divisible by ( x − 3 ) , we set p ( 3 ) equal to zero. Solving the equation reveals that the value of c must be 4.
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