When it comes to the mathematics portion of standardized tests there are at least two F-words that the test writers are actually hoping you’ll think of: “FOIL” and “factor”. In this post, we’ll take a look at these two important polynomial techniques and why they’re so useful.
What are polynomials?
Polynomials are algebraic terms that include a constant multiplied by a variable that’s been raised to an exponent. That means that when you see an algebraic expression on the SAT in the form of axn, you’re probably dealing with a polynomial.
Parts of a Polynomial Term
a = coefficient (a real number, like 6, 5.2, -3, etc.)
x = variable
n = exponent ≥ 0
Polynomials can contain multiple terms, and even multiple variables. For instance, the following are all examples of polynomials:
- 3x3 + 2x2 + 4x + 13
- x2 + 3xy – y
- 18x4 + 2x + 9
On the other hand, there are a few things that polynomials never include:
- no negative exponents
- no variables as exponents
The highest exponent in the expression tells us what degree a polynomial is. For example, expression number 1 above is considered a “third degree” polynomial because it’s greatest exponent is a 3. Expression number two is considered a “second degree” polynomial because it’s greatest exponent is a 2, while expression number 3 is considered a “fourth degree” polynomial, because it’s greatest exponent is 4.
Factoring Polynomials
Seemingly complex polynomials are a common, and potentially intimidating, sight on the SAT. Just looking at a long string of algebraic terms under the high-stress conditions of a standardized test can be daunting! Thankfully, most lengthy polynomials that appear on the SAT can be simplified into something more manageable using a technique known as factoring.
To simplify a polynomial, begin by looking for a “factor” that all of the terms have in common. For example, if you had the expression 3x2 +3x, you could identify 3x as a factor that both terms can be divided by and which can hence be pulled out as a multiplier: 3x2 +3x → 3x(x + 1). As another example, by factoring we can simplify 8x3 +12x2 +4x to 4x(2x2 +3x + 1).
Factors aren’t limited to being a single term, however. For example, factors may include multiple variables (i.e. 3x2y2 + x2y + xy2 factors to xy(3xy + x + y)).
Even more complicated, factors can actually contain multiple terms. Identifying two-term factors (two-term polynomials are also known as binomials) can be difficult since it involves thinking through how both terms in the binomial would multiply by each of the other terms in the expression. When correctly identified, however, binomial factors can greatly simplify otherwise unwieldy equations. For example, through factoring an expression like 18x2+ 38x + 12 can be made into the much more manageable expression (9x+3)(2x + 4).
FOILing
While it may sound like something done to a plate of leftovers, FOILing is actually a technique used to expand expressions involving two binomials into a single polynomial expression and at heart is the opposite of factoring. Because each term in each binomial must be multiplied by each term in the other binomial, it can sometimes be confusing trying to keep track of all of the terms. The letters in the word FOIL are a helpful acronym to remember how to correctly multiply binomials without missing anything:
F = First
O = Outside
I = Inside
L = Last
To foil a generic expression like (ax + b)(cy + d) you would follow the FOIL method this way:
F = First (multiply the first terms in each of the binomials)= ax * cy
O = Outside (multiply the term on the far left of first polynomial by the term on the far right of the second polynomial) = ax * d
I = Inside (multiply the term on the right side of the first polynomial by the term on the left side of the second polynomial) = b * cy
L = Last (multiply th second term in each of the binomials) = b * d
By using the FOIL approach, we have made sure that each term has been appropriately multiplied by all of the other terms. Adding them together we get the result:
axcy + adx + bcy * bd
The following is an example using real terms:
(7x + 3)(4x + 2)
F = 7x * 4x = 28x2
O = 7x * 2 = 14x
I = 3 * 4x = 12x
L =3 *2 = 6
When we add these terms up the result is the second-degree polynomial 28x2 + 26x + 6.
Because of their frequent appearance in real-life math and science, learning how to handle polynomials is an important part of preparing not only for standardized tests, but for college-level learning. Being able to factor and FOIL will significantly simplify your encounters with polynomials on these exams and increase your ability to obtain meaningful information from otherwise potentially unwieldy mathematical expressions.