If $2 n^3-7 n^2+n+10=(n-1) Q(n)+R$, find the remainder $R$ and quotient $Q(x)$.
Answer (2)
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 .
Perform synthetic division to find the coefficients of the quotient.
Determine the quotient Q ( n ) from the synthetic division result: Q ( n ) = 2 n 2 − 5 n − 4 .
State the remainder and the quotient: R = 6 and Q ( n ) = 2 n 2 − 5 n − 4 . 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 the quotient Q ( n ) and the remainder 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, we are dividing by ( n − 1 ) , so c = 1 . We substitute n = 1 into the polynomial 2 n 3 − 7 n 2 + n + 10 to find R .
Calculating the Remainder Substituting n = 1 into the polynomial, 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. Let's use synthetic division. We divide 2 n 3 − 7 n 2 + n + 10 by ( n − 1 ) .
Performing Synthetic Division Using synthetic division:
1 | 2 -7 1 10
| 2 -5 -4
------------------
2 -5 -4 6
The numbers in the bottom row (excluding the last number) are the coefficients of the quotient Q ( n ) . The last number is the remainder R , which we already found to be 6.
Determining the Quotient The quotient Q ( n ) is a polynomial of degree 2 (one less than the original polynomial). The coefficients are 2, -5, and -4. Therefore, the quotient is Q ( n ) = 2 n 2 − 5 n − 4 .
Final Answer Thus, the remainder is R = 6 and the quotient is Q ( n ) = 2 n 2 − 5 n − 4 .
Examples
Polynomial division is a fundamental concept in algebra and has practical applications in various fields. For example, engineers use polynomial division to analyze and design control systems. Suppose an engineer models a system's behavior using a polynomial equation. By dividing this polynomial by another related polynomial, they can simplify the model, making it easier to understand and control the system's response. This simplification helps in designing stable and efficient control mechanisms.
The quotient Q ( n ) from the division of the polynomial 2 n 3 − 7 n 2 + n + 10 by n − 1 is 2 n 2 − 5 n − 4 , and the remainder R is 6 .
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