Below is a list of some ``simple'' algebra problems. Some
of the solutions are correct and some of them are {wrong}! For each
problem:
- determine if the answer is correct;
- determine if there are any mistakes made in solving the problem and list
them ({note} that just because the answer is correct does not mean there
are no mistakes);
- if the answer to (A) and/or (B) is {NO}, redo the problem
correctly; if the answers to (A) and (B) are {YES}, devise another
correct method to solve the problem.
\begin{align*}
a) &\frac{x^2-1 }{ x+1}=\frac{x^2 + (-1) }{ x+1}=\frac{x^2 }{ x} + \frac{-1}{ 1}=x-1
\\
b) & (x+y)^2 - (x-y)^2=x^2+y^2-x^2-y^2=0\\
c) &\frac{9(x-4)^2 }{ 3x-12}= \frac{3^2(x-4)^2 }{ 3x-12}=\frac{(3x-12)^2 }{
3x-12}=3x-12 \\
d) & \frac{x^2y^5}{ 2x^{{-3}}}=x^2y^5\cdot
2x^3=2x^6y^5 \\
e) & \frac{(2x^3+ 7x^2 + 6)-(2x^3-3x^2-17x+3) }{ (x+8) + (x-8)}\\ &=
\frac{(2x)^2-17x+9 }{ 2x}=2x-17x+9=-15x+9=-6x \\
f) &\frac{x^{{-1}}+ y^{{-1}} }{ x^{{-1}} - y^{{-1}}}=\frac{(x+y)^{{-1}}
}{ (x-y)^{{-1}}}=(\frac{x+y }{ x-y})^{{-1}}=- \frac{x+y }{ x-y}=\frac{x+y
}{ y-x}
\end{align*}
Tags: algebra, rational, higher-polynomial, illustrative-example, computation, true-false, a--a, t1, c1
If you are designing a worksheet to review some of the basic rules of algebra this is a great worksheet problem to include. It focuses on common mistakes/misconceptions made by students during an algebra course. In addition, with slight modification, this problem could also be incorporated in a worksheet designed for an algebra course. The times that I have used this problem in my discussion sections, it went over very well.
– Darrin Visarraga 1998-08-09