Andy divides $\left(2 x^4-3 x^3-3 x^2+7 x-3\right)$ by $\left(x^2-2 x+1\right)$ as shown below. What error does Randy make?
$\begin{array}{r}
2 x^2+x+3 \\
x^2-2 x+1{\overline{\smash{\big)}\left)2 x^4-3 x^3-3 x^2+7 x-3\right.}} \\
\frac{2 x^4-4 x^3+2 x^2}{x^3-5 x^2+7 x} \\
\frac{x^3-2 x^2+x}{-3 x^2+6 x-3} \\
\frac{-3 x^2+6 x-3}{0}
\end{array}$
A. He makes a subtraction error.
B. He makes an error writing the constant term in the quotient.
C. He makes an error choosing the $x$-term in the quotient.
Answer (1)
Perform polynomial long division of ( 2 x 4 − 3 x 3 − 3 x 2 + 7 x − 3 ) by ( x 2 − 2 x + 1 ) .
The correct quotient is 2 x 2 + x − 3 .
Compare the correct quotient with Randy's quotient 2 x 2 + x + 3 .
Randy made an error writing the constant term in the quotient: He mak es an error w r i t in g t h eco n s t an tt er min t h e q u o t i e n t . .
Explanation
Problem Analysis We are given a polynomial long division problem and asked to identify the error made by Randy. The division is of ( 2 x 4 − 3 x 3 − 3 x 2 + 7 x − 3 ) by ( x 2 − 2 x + 1 ) . Randy's result is a quotient of 2 x 2 + x + 3 with a remainder of 0. We need to perform the long division ourselves to find the correct quotient and identify the error.
Performing Long Division Let's perform the polynomial long division step by step:
Divide 2 x 4 by x 2 to get 2 x 2 . Multiply 2 x 2 by ( x 2 − 2 x + 1 ) to get 2 x 4 − 4 x 3 + 2 x 2 . Subtract this from ( 2 x 4 − 3 x 3 − 3 x 2 ) to get x 3 − 5 x 2 . Bring down the 7 x term to get x 3 − 5 x 2 + 7 x .
2 x 2 x 2 − 2 x + 1 ) 2 x 4 − 3 x 3 − 3 x 2 + 7 x − 3 − ( 2 x 4 − 4 x 3 + 2 x 2 ) x 3 − 5 x 2 + 7 x
Divide x 3 by x 2 to get x . Multiply x by ( x 2 − 2 x + 1 ) to get x 3 − 2 x 2 + x . Subtract this from ( x 3 − 5 x 2 + 7 x ) to get − 3 x 2 + 6 x . Bring down the − 3 term to get − 3 x 2 + 6 x − 3 .
2 x 2 + x x 2 − 2 x + 1 ) \2 x 4 − 3 x 3 − 3 x 2 + 7 x − 3 − ( 2 x 4 − 4 x 3 + 2 x 2 ) x 3 − 5 x 2 + 7 x − ( x 3 − 2 x 2 + x ) − 3 x 2 + 6 x − 3
Divide − 3 x 2 by x 2 to get − 3 . Multiply − 3 by ( x 2 − 2 x + 1 ) to get − 3 x 2 + 6 x − 3 . Subtract this from ( − 3 x 2 + 6 x − 3 ) to get 0.
2 x 2 + x − 3 x 2 − 2 x + 1 ) \2 x 4 − 3 x 3 − 3 x 2 + 7 x − 3 − ( 2 x 4 − 4 x 3 + 2 x 2 ) x 3 − 5 x 2 + 7 x − ( x 3 − 2 x 2 + x ) − 3 x 2 + 6 x − 3 − ( − 3 x 2 + 6 x − 3 ) 0
The correct quotient is 2 x 2 + x − 3 .
Identifying the Error Comparing the correct quotient 2 x 2 + x − 3 with Randy's quotient 2 x 2 + x + 3 , we see that the error is in the constant term. Randy has + 3 instead of − 3 .
Conclusion Therefore, Randy made an error writing the constant term in the quotient.
Examples
Polynomial long division is used in various applications, such as simplifying complex rational expressions in calculus, designing control systems in engineering, and decoding error-correcting codes in data transmission. For instance, when analyzing the stability of a control system, engineers often need to simplify transfer functions, which are rational expressions. Polynomial long division helps break down these functions into simpler terms, making it easier to analyze the system's behavior and design appropriate controllers.
