What is the remainder when $(x^4+36)$ is divided by $(x^2-8)$?
Answer (1)
Express the division as x 4 + 36 = ( x 2 − 8 ) q ( x ) + a x + b .
Substitute x 2 = 8 into the equation to get 64 + 36 = ( 8 − 8 ) q ( x ) + a x + b , which simplifies to 100 = a x + b .
Solve for a and b by using x = ± 2 2 , which gives a = 0 and b = 100 .
The remainder is therefore 100 .
Explanation
Problem Setup We are given the polynomial x 4 + 36 and asked to find the remainder when it is divided by x 2 − 8 .
Expressing the Division We can express the division as x 4 + 36 = ( x 2 − 8 ) q ( x ) + r ( x ) , where q ( x ) is the quotient and r ( x ) is the remainder. Since we are dividing by a quadratic, the remainder r ( x ) will be of the form a x + b , where a and b are constants.
Finding the Remainder Thus, we have x 4 + 36 = ( x 2 − 8 ) q ( x ) + a x + b . We can perform polynomial long division to find the quotient and remainder. Alternatively, we can use the fact that if x 2 = 8 , then x 2 − 8 = 0 .
Substitution If x 2 = 8 , then x 4 = ( x 2 ) 2 = 8 2 = 64 . Substituting x 2 = 8 into the equation x 4 + 36 = ( x 2 − 8 ) q ( x ) + a x + b , we get 64 + 36 = ( 8 − 8 ) q ( x ) + a x + b , which simplifies to 100 = a x + b .
Solving for a and b Since x 2 = 8 , x = ± 2 2 . If x = 2 2 , then 2 2 a + b = 100 . If x = − 2 2 , then − 2 2 a + b = 100 . Adding the two equations gives 2 b = 200 , so b = 100 . Subtracting the two equations gives 4 2 a = 0 , so a = 0 .
The Remainder Therefore, the remainder is 0 x + 100 = 100 . We can verify this by performing polynomial long division: x 4 + 36 = ( x 2 − 8 ) ( x 2 + 8 ) + 100 . So the remainder is indeed 100.
Final Answer The remainder when x 4 + 36 is divided by x 2 − 8 is 100.
Examples
Polynomial division is a fundamental concept in algebra and has practical applications in various fields. For instance, in engineering, polynomial division can be used to analyze the stability of systems. Consider a system described by a transfer function, which is a rational function (a ratio of two polynomials). By performing polynomial division, engineers can simplify the transfer function and analyze the system's behavior more easily. For example, if the transfer function is x 2 − 8 x 4 + 36 , finding the remainder helps in understanding the system's response to certain inputs, ensuring the system operates within desired parameters and avoids instability. This ensures designs are safe and effective.
