Use the remainder theorem to find [tex]$k=2 ; P(x)=x^3+5 x^2+2 x-5$[/tex]
Answer (2)
By applying the Remainder Theorem to the polynomial P ( x ) = x 3 + 5 x 2 + 2 x − 5 with k = 2 , we evaluate P ( 2 ) and find that the remainder is 27. This is obtained by substituting 2 into the polynomial and performing the calculations, leading to a final result of 27. Thus, the answer is 27 .
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Substitute x = 2 into the polynomial P ( x ) = x 3 + 5 x 2 + 2 x − 5 .
Calculate P ( 2 ) = ( 2 ) 3 + 5 ( 2 ) 2 + 2 ( 2 ) − 5 .
Simplify the expression: P ( 2 ) = 8 + 20 + 4 − 5 .
Find the remainder: P ( 2 ) = 27 , so the remainder is 27 .
Explanation
Understanding the Problem We are given the polynomial P ( x ) = x 3 + 5 x 2 + 2 x − 5 and we want to find the remainder when P ( x ) is divided by x − k , where k = 2 . The Remainder Theorem tells us that the remainder is simply P ( k ) .
Applying the Remainder Theorem To find the remainder, we need to evaluate P ( 2 ) . We substitute x = 2 into the polynomial: P ( 2 ) = ( 2 ) 3 + 5 ( 2 ) 2 + 2 ( 2 ) − 5
Calculating P(2) Now, let's calculate each term: ( 2 ) 3 = 8 5 ( 2 ) 2 = 5 ( 4 ) = 20 2 ( 2 ) = 4 So, we have: P ( 2 ) = 8 + 20 + 4 − 5
Finding the Remainder Adding the terms together: P ( 2 ) = 8 + 20 + 4 − 5 = 32 − 5 = 27 Therefore, the remainder when P ( x ) is divided by x − 2 is 27.
Examples
The Remainder Theorem is useful in various applications, such as determining if a number is a root of a polynomial or simplifying complex polynomial expressions. For example, in engineering, when analyzing systems modeled by polynomials, the Remainder Theorem can help quickly assess the system's behavior at specific input values. It also has applications in coding theory and cryptography, where polynomials are used to encode and decode information.
