We can multiply any two algebraic expressions by multiplying all the terms in the first expression by all the terms in the second expression.
Example:
5x * 7y = 35xy
(5a + 2)(4b + 9) = 20ab + 45a + 8b + 18
We can divide any algebraic expression by another algebraic expression when they share factors. (See the discussion on factoring algebraic expressions.)
Example:
26xy/13x = 2y
When multiplying two algebraic expressions on the SAT, we can often use the “FOIL” technique. “FOIL” stands for “First, Outer, Inner, Last,” and refers to the order in which the terms of the two expressions are multiplied by one another.
You have probably used FOIL in your math classes, but if you used some other technique there’s no need to worry.
Example:
To multiply the expressions (5x + 7) and
(3x + 4), we can use FOIL.
The “First” pair in the acronym is the 5x and the 3x, because they are the first terms in each expression. We multiply these and get 15x2.
The “Outer” pair in the acronym is the 5x and the 4. We multiply these and get 20x.
The “Inner” pair in the acronym is the 7 and the 3x. We multiply these and get 21x.
The “Last” pair in the acronym is the 7 and the 4. We multiply these and get 28.
Now we just add up all those terms and we get the expression 15x2 + 20x + 21x + 28, which we can simplify a little bit by combining the two like x terms, giving us:
15x2 + 41x + 28
So (5x + 7)(3x + 4) = 15x2 + 41x + 28
If this seems a little complicated now, don’t worry about it. You’ll get it with practice—and there isn’t that much of it on the SAT anyway.
Factoring algebraic expressions
On the SAT, factoring an algebraic expression involves breaking the expression down into two other expressions that could be multiplied by each other to give the original expression.
Example:
If we have an algebraic expression like (8x + 4), we can break that down into the factors 4 and (2x + 1), because 4(2x + 1) = 8x + 4.
On the SAT, there are three types of factoring situations you’ll need to recognize:
orecognizing common factors
odoing “FOIL” in reverse
orecognizing a difference of squares
Recognizing common factors involves noticing that every term in a given expression has a common factor, as we did in the last example.
Example:
In the expression (21xy + 7x), both of the terms in the expression have a common factor of 7x, so we can factor the expression like this: 7x(3y + 1).
Factoring polynomials basically involves doing the “FOIL” process in reverse. Trust me, it’s not as hard as it looks. It just takes a little practice.
Example:
9x2 – 21x + 12 = (3x – 3)(3x – 4)
5x2 – 3x – 2 = (5x + 2)(x – 1)
When we factor the difference of two squares, there’s a shortcut we can use—the difference of two squares can be factored as the product of the sum of the square roots of the two squares times the difference of the square roots of the two squares. Let’s see an example.
Example:
9x2 – 4 = (3x + 2)(3x – 2)
Exponents
An exponent of a number is what we get when we multiply the number by itself a certain number of times.
Example:
x * x * x = x3 is an example of an exponential expression. The 3 in this example is the exponent, and the x is called the “base.”
Exponents can be positive or negative.
When an exponent is positive, we multiply the base by itself as many times as the exponent indicates, just like we did in the above example.
When an exponent is negative, we treat it just like a positive exponent EXCEPT that we take the reciprocal of the final amount (take another look at the discussion of reciprocals on page).
Example:
x5 = x * x * x * x * x
x -5 = 1/(x5)
We can multiply exponent expressions by each other when the bases are identical. To do that, we just add the exponents:
Example:
(x6)(x4) = (x * x * x * x * x * x) (x * x * x * x) =
x * x * x* x * x * x * x * x * x * x = x10
(x7)(x - 4) = x3
We can also divide exponent expressions when they have the same base. For that we just subtract the exponents:
Example:
(x8)/(x2) = x6
Finally, we can raise exponential expressions to other exponents by multiplying the first exponent by the second one:
Example:
(x4)5 = x20
Note that raising any number to an exponent of zero gives you the number 1.
Example:
y0 = 1
Using equations
On the SAT, an equation is a statement that involves an algebraic expression and an equals sign.
Example:
5x = 20 is an equation, because it involves the algebraic expression 5x and an equals sign.
Solving an equation means figuring out how much the variable in the equation is worth. We can solve equations just like you learned in algebra class—by multiply, dividing, adding, or subtracting both sides of the equation by the same amounts until we’re left with a value for the variable.