Algebra Pitfalls

The following page contains a list of common errors committed by students of Calculus.

IMPORTANT: ANYTHING WRITTEN IN MAROON IS FALSE! Maroon text represents common mistakes that Calculus students must avoid.

MYTH: (x + y)^2 = x^2 + y^2 ; sqrt(x + y) = sqrt(x) + sqrt(y) ; in general (x+y)^n = x^n + y^n

COUNTEREXAMPLE: Suppose x = 9 and y = 16. Then we have

So, clearly, in many cases,

So, clearly, in many cases,

WHY IT IS WRONG: If you recall your PEMDAS rules from early algebra, this may make more sense. When we simplify algebraic expressions, we have to follow a certain order of steps. We evaluate expressions inside parentheses first. Because of that, we can't distribute a power within the sum - we have to do the sum first.

If this is confusing, just try it out for yourself. Using the rules of multiplying polynomials, we can test to see that (x + y)²≠ x² + y².

(x + y)² = (x + y)(x + y) = x(x + y) + y(x + y) = x² + xy + xy + y²
(x + y)² = x² + 2xy + y² ≠ x² + y²

In conclusion, do not distribute exponents or roots across sums. You either have to multiply out the exponent, or try to factor the expression inside the root or the exponent.

NOTE: You may distribute an exponent across a product, or a division. For example, (ab)ⁿ = aⁿbⁿ, and (a/b)ⁿ = aⁿ/bⁿ. For an explanation, check out the page on exponential rules.

MYTH: or ; i.e. partially cancelling fractions

COUNTEREXAMPLE: Suppose x = 2. Then we know that

but , and .
Note how the expressions are not equal.

WHY IT IS WRONG: When we have a fraction with a sum in the numerator, the denominator is dividing the entire numerator. In our example, we cancelled the x with only a single term - effectively we divided just the 3x term by x. This is a problem - we need to divide every part of the numerator by the entire denominator.

The numerator, x² + 3x + 4, is divided by the denominator, x. That means that each term of the numerator, x², 3x, and 4, is divided by the denomator. So

In conclusion, you can distribute a denominator to an entire sum. The key is that, in order to cancel a term from the denominator, it must be divided out of each term in the numerator. In general, if you can't cancel out a term from each part of the numerator, just leave the fraction as is - it probably won't help you that much.

NOTE: When you have a product that is being divided, you actually only divide a single term by the denominator. For example, if we have the product aⁿbⁿ divided by a, we can just divide the aⁿ part. This is because division and multiplication are equivalent actions, so they occur concurrently. With our current myth, division and addition are different order actions, so we divide the entire sum.