If $2 n^3-7 n^2+n+10=(n-1) Q(n)+R$, find the remainder $R$ and quotient $Q(n)$.
Answer (2)
The quotient of the polynomial 2 n 3 − 7 n 2 + n + 10 when divided by n − 1 is Q ( n ) = 2 n 2 − 5 n − 4 , and the remainder is R = 6 .
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Use the Remainder Theorem to find the remainder by substituting n = 1 into the polynomial: R = 2 ( 1 ) 3 − 7 ( 1 ) 2 + 1 + 10 = 6 .
Assume the quotient is a quadratic polynomial Q ( n ) = a n 2 + bn + c and substitute it into the given equation.
Compare the coefficients of the powers of n to find the values of a , b , and c .
Determine the quotient Q ( n ) = 2 n 2 − 5 n − 4 and the remainder R = 6 , so the final answer is Q ( n ) = 2 n 2 − 5 n − 4 ; R = 6 .
Explanation
Problem Analysis We are given the polynomial equation 2 n 3 − 7 n 2 + n + 10 = ( n − 1 ) Q ( n ) + R , where Q ( n ) is the quotient and R is the remainder when the polynomial 2 n 3 − 7 n 2 + n + 10 is divided by ( n − 1 ) . Our goal is to find Q ( n ) and R .
Finding the Remainder We can use the Remainder Theorem to find the remainder R . The Remainder Theorem states that if we divide a polynomial P ( n ) by ( n − c ) , the remainder is P ( c ) . In this case, c = 1 . Substituting n = 1 into the polynomial 2 n 3 − 7 n 2 + n + 10 , we get:
R = 2 ( 1 ) 3 − 7 ( 1 ) 2 + ( 1 ) + 10 = 2 − 7 + 1 + 10 = 6.
So, the remainder R is 6.
Finding the Quotient Now, we need to find the quotient Q ( n ) . We can perform polynomial long division or synthetic division. Here, we'll deduce it. Since the dividend is a cubic polynomial and the divisor is a linear polynomial, the quotient will be a quadratic polynomial of the form Q ( n ) = a n 2 + bn + c .
We have 2 n 3 − 7 n 2 + n + 10 = ( n − 1 ) Q ( n ) + 6 . Substituting Q ( n ) = a n 2 + bn + c , we get:
2 n 3 − 7 n 2 + n + 10 = ( n − 1 ) ( a n 2 + bn + c ) + 6 2 n 3 − 7 n 2 + n + 10 = a n 3 + b n 2 + c n − a n 2 − bn − c + 6 2 n 3 − 7 n 2 + n + 10 = a n 3 + ( b − a ) n 2 + ( c − b ) n − c + 6
Comparing the coefficients of the powers of n , we have:
n 3 : a = 2 n 2 : b − a = − 7 ⇒ b = a − 7 = 2 − 7 = − 5 n 1 : c − b = 1 ⇒ c = b + 1 = − 5 + 1 = − 4 Constant: − c + 6 = 10 ⇒ − c = 4 ⇒ c = − 4
Thus, Q ( n ) = 2 n 2 − 5 n − 4 .
Final Answer Therefore, the quotient is Q ( n ) = 2 n 2 − 5 n − 4 and the remainder is R = 6 .
Examples
Polynomial division is a fundamental concept in algebra and has practical applications in various fields. For example, in engineering, polynomial division can be used to analyze the stability of systems. Imagine a control system where the characteristic equation is a polynomial. By dividing this polynomial, engineers can determine the system's response and stability, ensuring it operates safely and efficiently. This helps in designing robust and reliable control systems for various applications.
