A polynomial is a collection of terms joined by and operators. Polynomials with one term are called monomials, with two terms is called binomial, with three terms is a trinomial, and so on.
Another (and more important) way to classify a polynomial is by its degree. The degree is the value of the highest exponent (power) in the polynomial. For example:
is a second-degree polynomial. Similarly,
is a seventh-degree polynomial.
Finally, each term can be “named” by its exponent. If a term has the exponent 2, it is called the “square” term, exponent 3 is the “cubic” term, exponent 4, the “quartic” term, and so on.
Next we will cover multiplying two binomials.
Got all that? Let’s go…
Multiplying binomials is taught with an acronymn — FOIL — first, outer, inner, last. How does this work?
- The first terms: are and so their product is
- The outer terms: are and so
- The inner terms: are and so
- The last terms: are and so
Now, sum them all together into the expanded polynomial:
Let’s try some examples.
- Using the FOIL technique, multiply out
- Use FOIL to multiply
This is a place where students get tripped up. Note that
Here’s the correct way:
Now, FOIL it out, giving:
You just saw the first special case:
Next, learn this one:
The terms cancel each other, leaving:
So, remember this: the difference of two squares can be factored into
Squares and Square Roots
Here’s a good place to mention this. Question: What is ? Immediate answer: 2 !
But that’s not correct. What? The correct answer: Why? Because also.
When you take a square root, always think “”.
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