- Now we're moving into polynomials, one of the most important building blocks in SAT advanced math. Quadratics, factoring, graphing functions, and many of the algebra topics you'll see later all build on the ideas in this lesson.

Let's start with the definition. A polynomial is a sum of terms, and each term contains a variable raised to a non-negative integer exponent. That phrase, non-negative integer exponent, is the key rule for this entire lesson.

If you understand that rule, you can quickly determine whether an expression is a polynomial or not. Let's look at the parts of a polynomial. On the screen, we've got 4X cubed plus 2X squared minus 5X plus seven. Each piece separated by a plus or minus sign is called a term.

The numbers attached to the variables are called coefficients. And the number with no variable attached is called the constant term. One important detail? In negative 5X, the coefficient is negative five.

The sign travels with the coefficient. Another important vocabulary word is degree. For a single term, the degree is simply the exponent. And for the entire polynomial, the degree is the highest exponent that appears. In this example, the highest exponent is three, so the polynomial has a degree of three.

One more term you'll see frequently, leading term. The leading term is the term with the highest degree, and it's coefficient is called the leading coefficient. In our example, 4X cubed is the leading term, and four is the leading coefficient. Now, let's come back to the exponent rule.

For an expression to count as a polynomial, every variable exponent must be a non-negative integer. That means no negatives, no fractions, no decimals. Just zero, one, two, three, and so on. And this is where our previous lessons connect. A square root can be written as X raised to the power of one half.

A variable in the denominator can be written as X raised to a negative power of one. Both violate the exponent rule, so neither one is a polynomial. The exponent rule explain what does and does not qualify. And while it might be nice to know conceptually what is and isn't a polynomial, you certainly aren't going to be tested on whether you can identify one.

So don't fret if memorizing terminology isn't your strong suit. Ergo, we are not going to spend any more time here. Let's move on and quickly talk about classification by terms. One term is a monomial, two terms is a binomial, three terms trinomial, four or more terms, we usually just group all of that together and call them all polynomials.

These names appear fairly often in SAT question wording, so it's helpful to be somewhat familiar with them. Another way you might see a polynomial classified is by degree. To the first degree is linear, to the second degree, a quadratic, and to the third degree is a cubic.

Familiarity with quadratics is especially important because they appear frequently throughout SAT advanced math. All right, now let's talk about standard form. Standard form means writing terms in descending order by degree. Highest exponent first, then the next highest, and so on, until a constant term.

This is the format you'll see throughout SAT questions and answer choices. Here's an example. Five plus 2X cubed minus four X squared. That's not written in standard form. To rewrite it, place the highest degree term first, 2X cubed, then the second highest degree minus 4X squared, then the constant plus five.

One important reminder, the sign travels with the terms when you move them. Don't leave a negative behind. Before we add and subtract polynomials, we need one more idea, like terms. Two terms are like terms only if they have the same variable and the same exponent.

Both conditions must be true. For example, 3X squared and 5X squared. Those are like terms. We can combine them into 8X squared. But 3X squared and 5X cubed are not like terms. Different exponents means the terms cannot be combined.

That gives us the rule for adding and subtracting polynomials. Only like terms can be combined. Everything else stays exactly where it is. Okey-dokey, let's talk about adding polynomials. It's usually a straightforward process.

Step one, combine like terms. Step two, write the result in standard form, and that is the entire process. Let's try one. 3X squared plus 5X plus two plus X squared minus 4X plus seven.

Let's combine the X squared terms, X terms, and constants. Doing so is going to give us 3X squared plus X squared plus 5X minus 4X plus two plus seven. And once we do the addition, we get 4X squared plus X plus nine, which is already in standard form, so we're done.

Now, subtracting is pretty much the same deal, however, with subtracting, there's a catch. When a minus sign appears in front of the parentheses, it distributes to every term inside, so every sign changes. Take a look at the visual.

On the left, the correct distribution. On the right, the classic mistake. When rushing, students sometimes flip the first sign and forget the rest. Let's run the full example. First, distribute the negative all the way through, and then combine like terms.

And once we simplify, we have 3x squared minus 10x plus 10. And that's as simplified as this can go, so this one is done. Now it's your turn. Pause the video and try these five questions.

Take your time and resume when ready. Now, take a moment to check the answers. If you missed any of these, go back and revisit the associated rule. All right, everybody, it's time for the summary. A polynomial contains only non-negative integer exponents.

A few key vocabulary items are term, coefficient, constant, degree, and leading term. But knowing these by name isn't nearly as important as knowing how to navigate them. The SAT tests your ability to manipulate polynomials, not whether you know what the various parts and pieces are called.

Standard form is written in descending power order, that's the format SAT answer choices will take. When simplifying, only combine like terms. And finally, be careful with subtraction, especially if you're pressed for time and you have one of those long polynomials.

Be sure you fully distribute the negative before combining. And that my friends is going to do it for polynomial basics. Thanks for watching.