The SBC GMAT Files
Simplifying Algebraic Expressions
Staff, Test Prep New York/Test Prep San Francisco
In math, essentially the same information can be expressed in multiple ways, and you will often have to change such expressions from one form to another in order to answer GMAT questions. The most common manipulations you will need on the test are factoring and expanding algebraic expressions. So let’s start with the basic rules and conclude with some shortcuts and examples. You might know some of this material already; in fact, you should know it if you did well in your high school and college math courses, but be sure to look through this post to refresh these skills as you will certainly need them on the test.
The Distributive Property
When a group of numbers in parentheses is multiplied by a number, each number in the parentheses is multiplied by that number.
x(y + z) can be written as xy + xz
x(y ”“ z) can be written as xy ”“ xz.
Remember that whenever you multiply the expression in the brackets by a negative number, you must also distribute a negative sign:
-x(y + z)= -xy ”“ xz
The reverse can be done as well. When you see a sum of two numbers or expressions that contain a common factor, you can factor out that common factor.
In the expression xy+xz, both terms contain x, so we can factor it out like so: x (y+z).
If you see an expression in which two binomials are multiplied, for example (a + b)(c + d), you will most likely need to expand or FOIL them., FOIL is an abbreviation for First, Outer, Inner, Last. Basically, this technique explains in which order you multiply terms of the two binomials that you have to expand. Let’s proceed with our example:
(a + b)(c + d) = ac + ad + bc +bd
Here ac are the First terms in each binomial; ad are the Outer terms in each binomial; bc are the Inner ones, and bd are the Last terms.
On the GMAT, expanding binomials often provides opportunities to cancel out some terms, or to create a quadratic equation. For example:
(a + 5)(a + 7) = a2 + 7a + 5a + 35 = a2 + 12a +35
Formulas of Short Multiplication
There are some special cases when expressions can be factored or distributed quickly by spotting certain tendencies. These tendencies are shortcuts, and are called formulas of short multiplication. They are often tested on the GMAT and even though you can factor or expand them using the common methods, spotting opportunities to apply these formulas will allow you to save time. And time is crucial on the GMAT. Here are the formulas:
(x + y)2 = x2 + 2xy + y2
(x – y)2 = x2 – 2xy + y2
x2 – y2 = (x + y)(x – y)
The opposite is also true, so when you see an expression in the form x2 + 2xy + y2 you can quickly factor it to get (x + y)2.
Let’s look at an example to see how these properties can be tested on the GMAT through a Data Sufficiency Question
m4 ”“ n4=
(1) m2 ”“ n2 = 36
(2) m + n = 10
(A) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.
The expression m4 ”“ n4 should remind you of the formula of short multiplication x2 – y2 = (x + y)(x – y), but here x = m2 and y = n2. So, m4 ”“ n4 = (m2 ”“ n2)(m2 + n2).
The first statement gives you the value of (m2 ”“ n2), but gives no information that would allow you to calculate the value of m2 + n2, and is therefore not sufficient on its own. We can eliminate choices A and D, as they assume sufficiency of the first statement.
NOTE: Anytime that ”˜A’ doesn’t work, you can also eliminate ”˜D’. Likewise, anytime ”˜B’ doesn’t work, you can eliminate ”˜D’. You can therefore set up on your erasable scratch pad the following set up: AD/BCE or BD/ACE, and move through the letters accordingly. After you eliminate A or B, then see if the following three choices are accurate.
No A No D, therefore, BCE
No B No D, therefore, ACE
The second statement gives us the value of m + n = 10. To get at least some use of this information we should further factor our expression:
m4 ”“ n4 = (m2 ”“ n2)(m2 + n2) = (m – n)(m + n)(m2 + n2) = 10(m – n)(m2 + n2).
Again, this statement only gives us the value of one factor, while leaving the other two unknown. We can eliminate choice B. Now we are left with choices C and E, and to determine which one is correct, we need to evaluate both statements together.
From statement (1) we know that m2 ”“ n2 = 36 or (m – n)(m + n)=36, we can substitute (m + n) with 10 from the second statement to get 10(m ”“ n) = 36. Dividing both sides by 10, we can determine that m ”“ n = 3.6.
Now we have two linear equations:
m ”“ n = 3.6
m + n = 10
You have a system of two independent linear equations, and can solve for the individual values of m and n, and knowing these values will certainly allow you to calculate the value of m4 ”“ n4. But since this is a Data Sufficiency question, you do not need to do any calculations. As soon as you know that together the statements can lead you to the answer, you can pick choice C.
(If this weren’t a data sufficiency question, you can set these up as an equation to get ONE variable so you can determine the answer:
m ”“ n = 3.6
+ m + n = 10
2m = 13.6
therefor m = 6.8, the you can figure out n = 3.2
The GMAT’s Favorite Question
GMAT likes to test your ability to recognize opportunities to factor and expand expressions. Whenever you see an expression that can be factored””factor it; when you see an expression that can be expanded””expand it. But the most frequently encountered quadratic expression on the GMAT is x2 ”“ 1. If you see an expression like this on the GMAT, keep in mind that 12 = 1, so the expression x2 ”“ 1 is the same as x2 ”“ 12 and can be factored into (x + 1)(x – 1). Remember sometimes they’ll manipulate the equation so that you don’t overtly recognize that it ”˜is’ some form of a quadratic equation. While this is a very simplified equation, notice how it’s an (x-y)2 :
Another thing to have in your back pocket (metaphorically) is Pascal’s Triangle. Knowing this a mini- trick to give students an edge, but it’s non-essential content.
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 5 1…
Pascals’ Triangle can be really useful for quickly determining the coefficients of binomial expansions, particularly in the case of (x + y)3. At the advanced level, GMAT wants to see that you can apply this to save time.
Pascal’s triangle determines the coefficients that are the result of binomial expansions.
For example, consider this equation:
(x + y)2 = x2 + 2xy + y2 = 1x2y0 + 2x1y1 + 1x0y2.
Notice the coefficients are the numbers in row two of Pascal’s triangle: 1, 2, 1.
Or this equation:
(x + y)4 = x4 + 4x3y + 6x2y2+ 4xy3 + y4
And so on.
Going over these properties and rules, and applying them on the GMAT, will allow you to be in great shape.
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