obstacle course

Part two of the essential guide to eliminating errors that trip up many test-takers.

 

Studying algebra is a key step in a student's mathematical journey. It’s where you start working with mathematical abstractions and key concepts like functions and graphs. It sets the foundation for later study in areas like calculus and statistics. Whether you are studying for the ACT or SAT, preparing for your pre-calculus class, or just learning algebra, it’s important to set yourself up for success.

 

In part one of “Common Algebra Mistakes”, we reviewed three areas where students are prone to make errors, including when using common mathematical symbols like equals and parentheses, when cancelling common factors, and in mistaking roots for solutions. In this second part, we explore linear function properties, misuse of the equals sign, exponents, polynomial fractions, and more. 

 

The best way to reduce errors in general is to develop a strong understanding of the underlying concepts. Think through why someone might make such a mistake, and the reasons the correct approach is, in fact, correct. That’s the approach of both this article and our excellent math tutors, who help students gain a rock-solid understanding of algebra, allowing them to excel in school and achieve higher scores on admissions tests such as the SAT and ACT.

1. Assuming the linear function property

Certain functions have a pair of properties that together we call “linearity”. Those properties are:

  • Additivity: f(x + y) = f(x) + f(y)
  • Homogeneity: f(ax) = af(x).

(Note that this is NOT the same as the functions of the form, which we call linear functions because their graphs are straight lines.)

 

Sometimes, students assume that all or some functions are linear functions and can be manipulated in this way when in reality they are not. Most notably, it is NOT generally the case that (x + y)^2 = x^2 + y^2 or that \sqrt{x + y} = \sqrt{x} + \sqrt{y}. Some simple computations make these cases clear:

  • By FOILING, we see (x + y)^2 = (x + y)(x + y) = x^2 + 2xy + y^2, which has an extra term when compared to x^2 + y^2, and so they are not equal
  • \sqrt{x + y}^2 = x + y, whereas (\sqrt{x} + \sqrt{y})^2 = x + 2\sqrt{xy} + y. We conclude that \sqrt{x + y}\not = \sqrt{x} + \sqrt{y} since they have different squares.

 

We could also check using specific numbers to see that these functions are not linear. For example, if x = y = 1, then (x + y)^2 = 4, whereas x^2 + y^2 = 2. So, we found a set of values where additivity doesn’t hold for g(x) = x^2 , and so the function is not additive in general. (CAREFUL: finding values where the properties don’t hold is enough to prove that the given functions don’t have the linearity property, but finding values where additivity and homogeneity do hold is NOT enough to show that the given functions have the linearity property. For example, in the cases we’ve already discussed, the equalities actually work when either x or y is zero.)

 

To tell for sure if a given function does have the linearity property, we have to prove it using algebraic manipulations. Let’s work through an example to demonstrate:
The simplest example is f(x) = x.

  • This function is additive: f(x + y) = x + y = f(x) + f(y)
  • It is also homogeneous: f(ax) = ax = af(x)

 

Here are some more common examples of functions that do NOT have the linearity property (see if you can prove it on your own!):

  • f(x) = e^x
  • h(x) = \frac{1}{x}
  • p(x) = ln(x)

2. "Equals" sign errors

Students can get very comfortable with “doing the same thing to both sides” when manipulating an equation. For example, if I had the equation \frac{x}{x+3} = \frac{7}{2x+6}, you might want to multiply both sides by 2x + 6 to get rid of the denominators. However, sometimes students can be overzealous with this tool and forget the real power and meaning of the equals sign.

 

For example, suppose you have a function f(x) = \frac{x}{x+3}. You might try to multiply both sides by x + 3 in order to get rid of the denominator on the right hand side. However, in this case, the equals sign is acting to define the function f, so multiplying on both sides won’t help you out much. You’d end up with (x + 3)f(x) = x, which will have you running in circles, since the only thing we know about f is that it is defined as \frac{x}{x+3} . You can’t just apply algebraic manipulation “to both sides” to “simplify” a function. It is defined in a certain way, and manipulating both sides of the equality in this case won’t help you see our function in a new light.

 

Similarly, when asked, for example, to simplify an expression like \frac{x+3}{x^2-9} , students might be inclined to try to get rid of the denominator by multiplying by x^2 - 9. The problem here is that we don’t have “two sides” because we don’t have an equation, just a stand-alone expression. We can manipulate both sides of an equals sign because we’re still keeping everything in balance. It’s like adding the same exact weight to two sides of a see-saw. However, if we don’t have an equation, we don’t have two quantities balancing each other, and so we can’t manipulate our expression in this way without fundamentally altering it.

3. Negative square roots and the square root symbol

Suppose you’re given the equation x^2 = 9 , and you’re trying to solve for x. Many students will try to solve this problem by taking the square root of both sides, and will find that x = 3. However, x = -3 is also a solution. Here we have to recognize the distinction between the meaning of the square root symbol (\sqrt{ }), and all of the true square roots of a number. By convention, the square root given by the expression \sqrt{x} of any (positive, real) number is a positive, real number. However, \sqrt{x} is not the only square root of x, where we define square root as any number that squares to x. Since (-1)^2 = 1, it will also be true that - \sqrt{x} is a square root of x. So, don’t forget those negative square roots when solving equations that involve squares!

4. All things exponents

Many students struggle with remembering exponent rules. When do you add them? When do you multiply them? What’s the difference between a fractional and a negative exponent? Here’s a quick refresher on the main rules relating to exponents:

  • By convention, x^0 = 1 , no matter what x is.
  • Raising a base to a power indicates how many times to multiply that base by itself. So, x^4 = xxxx. Remembering this fact specifically will help you remember all of the other rules. When in doubt, come back to this basic rule.
  • When multiplying terms with the same base, add the exponents: x^{a}x^{b} = x^{a+b} .
    • This makes sense when we refer back to the definition of the exponent. For example: x^{2}x^{3} = (xx)(xxx) = xxxxx = x^5 . Since the exponents mean how many times we multiply x by itself, and then we multiply those products together, overall we are multiplying x together by the number of times that is the sum of the two exponents.
  • When raising a term to a power, multiply exponents: (x^a)^b = x^{ab} .
    • Again, think about the definition. x^a inside the parentheses mean that we are going to multiply x by itself a times. But then, raising that to the b power means we are going to multiply x^a with itself, b times. So, overall, we get b copies of a copies of x, all multiplied together. Altogether that makes for a times b copies of x.
  • Negative exponents mean we are dividing by some number of copies of x. In particular, x^{-a} = \frac{1}{x^a} , and so we divide 1 by x a times. One way to remember this is to think about the addition rule. Since x^{a}x^{-a} = x^0 = 1, it must be that whatever x^{-a} is, it brings x^a back to 1, in essence “undoing” the a-times multiplication of x. Well, dividing by x(a times) would do the same thing, so it makes sense that negative exponents would indicate division.
  • A fractional exponent means we are taking a root. In particular, x^{1/a} = \sqrt[a]{x} . This makes sense when we think about the multiplication rule. We would need (x^{\frac{1}{a}})^a = x^{a/a} = x^1 = x. Since raising x^{1/a} to the a gives us x, it must be (essentially by definition) that x^{1/a} is the a-th root of x.

5. Cancelling polynomial fractions

It can be tricky to move from numerical fractions to polynomial fractions. In particular, many students make errors when it comes to cancelling terms in polynomial fractions. Consider the fraction \frac{2x+3}{2x+6} . Some students want to cancel the on the top and bottom. The problem with this is that we can only cancel common factors, so we cannot cancel individual terms in a sum. This makes more sense when we remember that division is the “mirror” or the “opposite” of multiplication. If we have a product in the numerator, we could “undo” that multiplication with a shared factor in the denominator, but without a shared factor, there’s nothing we can do.

 

Here’s an example of a correct sequence of computations:
\frac{x^2+5x+6}{x^2-9} = \frac{(x+2)(x+3)}{(x-3)(x+3)} = \frac{x+2}{x-3} .

Congratulations! You’ve made it through our guide to the common errors that trip up algebra students. But don’t worry if you feel a bit lost. Understanding algebra’s many twists and turns comes with plenty of “perfect practice”: not only working through problems, but reviewing any errors, understanding what happened when you made a mistake, and why the right answer is in fact right. 

 

You’re not alone in this. Our brilliant and caring math tutors have helped countless students nail middle and high school math and achieve high scores on admissions tests such as the SAT, ACT, SHSAT, SSAT and ISEE. Since math concepts build upon each other, the key to success is gaining a solid understanding at each math level, making it possible to move on to more advanced topics such as calculus with ease and confidence.

Meet Jenna

Jenna is a PhD candidate in mathematics at NYU. She recently graduated from Yale University, where she double majored in mathematics and philosophy. As an undergraduate, she cofounded a student group that works to alleviate barriers to access in STEM fields at Yale. As a tutor, she seeks to create a supportive and comfortable environment for students to flourish. Math can be a daunting subject for many; Jenna assists students by presenting mathematical ideas in an approachable and intuitive way.

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