Anja divides $\left(8 x^3-36 x^2+54 x-27\right)$ by $(2 x-3)$ as shown below. What error does Anja make?
$\begin{array}{l}
4 x^2-12 x+9 \\
2 x - 3 \longdiv { 8 x ^ { 3 } - 3 6 x ^ { 2 } + 5 4 x - 2 7 } \\
\frac{8 x^3-12 x^2}{-24 x^2+54 x} \\
\frac{-24 x^2+36 x}{18 x-27} \\
\frac{18 x-27}{0}
\end{array}$
A. She makes a subtraction error.
B. She makes an error choosing the $x$-term in the quotient.
C. She makes a multiplication error.
D. She makes an error rewriting the problem in long division.
Answer (1)
Perform polynomial long division of 8 x 3 − 36 x 2 + 54 x − 27 by 2 x − 3 .
Compare the result with Anja's steps.
Identify the multiplication error in the last step where Anja wrote 18 x + 27 instead of 18 x − 27 .
Conclude that Anja made a multiplication error. She makes a multiplication error.
Explanation
Problem Analysis We are given a polynomial long division problem and asked to identify the error Anja made. The problem is to divide 8 x 3 − 36 x 2 + 54 x − 27 by 2 x − 3 . We will perform the long division ourselves and compare it to Anja's work to find the mistake.
Performing Long Division Let's perform the polynomial long division:
Divide 8 x 3 by 2 x to get 4 x 2 . Multiply ( 2 x − 3 ) by 4 x 2 to get 8 x 3 − 12 x 2 . Subtract this from 8 x 3 − 36 x 2 to get − 24 x 2 .
Bring down the next term, + 54 x , to get − 24 x 2 + 54 x .
Divide − 24 x 2 by 2 x to get − 12 x . Multiply ( 2 x − 3 ) by − 12 x to get − 24 x 2 + 36 x . Subtract this from − 24 x 2 + 54 x to get 18 x .
Bring down the next term, − 27 , to get 18 x − 27 .
Divide 18 x by 2 x to get 9 . Multiply ( 2 x − 3 ) by 9 to get 18 x − 27 . Subtract this from 18 x − 27 to get 0 .
Correct Result So, the correct long division is:
4x^2 - 12x + 9
2x - 3 | 8x^3 - 36x^2 + 54x - 27
-(8x^3 - 12x^2)
------------------
-24x^2 + 54x
-(-24x^2 + 36x)
------------------
18x - 27
-(18x - 27)
----------
0
The quotient is 4 x 2 − 12 x + 9 and the remainder is 0 .
Identifying the Error Now let's compare this to Anja's work:
4 x 2 − 12 x + 9 2 x − 3 \longdiv 8 x 3 − 36 x 2 + 54 x − 27 − 24 x 2 + 54 x 8 x 3 − 12 x 2 18 x − 27 − 24 x 2 + 36 x − 54 18 x + 27
In the last step, Anja multiplies ( 2 x − 3 ) by 9 and gets 18 x + 27 , which is incorrect. It should be 18 x − 27 . Then, when subtracting ( 18 x − 27 ) from ( 18 x − 27 ) , the result should be 0 , not − 54 .
Conclusion The error Anja made is in the last step, where she incorrectly wrote 18 x + 27 instead of 18 x − 27 . This is a multiplication error.
Final Answer Therefore, the correct answer is: She makes a multiplication error.
Examples
Polynomial long division is a method used to divide one polynomial by another. It's similar to long division with numbers. For example, if you want to divide a plot of land represented by the polynomial x 3 + 3 x 2 + 5 x + 3 into sections each represented by x + 1 , polynomial long division helps determine the size and shape of the remaining land after division. This technique is also used in engineering to simplify complex transfer functions and analyze system stability.
