A polynomial is an expression that contains one or more algebraic terms, each consisting of a constant multiplied by a variable raised to a power greater than or equal to zero. For example, 
is a polynomial with three terms (the third term is 
. 
, on the other hand, is not a polynomial because x is raised to a negative power. A binomial is a polynomial with exactly two terms: 
and 
are both binomials.
The rest of this chapter will show you how to perform different operations on and with polynomials.
Multiplying Binomials
There is a very simple acronym that is useful in remembering how to multiply binomials. It is FOIL, and it stands for First, Outer, Inner, Last. This is the order that you multiply the terms of two binomials to get the right product.
For example, if asked to multiply the binomials:
You first multiply the first terms of each binomial:
Next, multiply the outer terms of the binomials:
Then, multiply the inner terms:
Finally, multiply the last terms:
Combine like terms and you have your product:
Here are a few more examples:
Multiplying Polynomials
Every once in a while, the Math IC test will ask you to multiply polynomials. It may seem like a daunting task. But when the process is broken down, multiplying polynomials requires nothing more than distribution and combining like terms.
Consider the polynomials (a + b + c) and (d + e + f). To find their product, just distribute the terms of the first polynomial into the second polynomial individually and combine like terms to formulate your final answer:
Here’s another example:
As you can see, multiplying polynomials is little more than rote multiplication and addition.
Quadratic Equations
A quadratic, or quadratic polynomial, is a polynomial of the form ax2 + bx + c, where a ≠ 0. The following polynomials are quadratics:
A quadratic equation sets a quadratic polynomial equal to zero. That is, a quadratic equation is an equation of the form ax2 + bx + c = 0. The values of x for which the equation holds are called the roots, or solutions, of the quadratic equation. Most of the questions on quadratic equations involve finding their roots.
There are two basic ways to find roots: by factoring and by using the quadratic fo-rmula. Factoring is faster, but it can’t always be done. The quadratic formula takes longer to work out, but it works for all quadratic equations. We’ll study both in detail.
Factoring
To factor a quadratic, you must express it as the product of two binomials. In essence, factoring a quadratic involves a reverse-FOIL process. Take a look at this quadratic:
In the example above, the leading term has a coefficient of 1 (since 1x2 is the same as x2). Since the two x variables are multiplied together during the FIRST step of foiling to get the first term of the quadratic polynomial, we know that the binomials whose product is this quadratic must be of the form (x + m)(x + n), where m and n are constants. You also know that the sum of m and n is 10, since the 10x is derived from multiplying the OUTER and INNER terms of the binomials and then adding the resulting terms together (10x = mx + nx, so m + n must equal 10). Finally, you know that the product of m and n equals 21, since 21 is the product of the two last terms of the binomials.
