The video examples below will demonstrate how to find enough information about a polynomial function to sketch its graph. Generally, the approach will be to

- Find the
**roots**, if possible- Greatest common factor (GCF), grouping, familiar pattern (quadratic, sum/difference of perfect cubes), rational root theorem

- Find the
**y-intercept, f(0)**, and any other easy-to-find points (usually just f(0) is necessary) - Determine the
**end behavior** - Remember that all polynomial functions have
**smooth, continuous graphs** - Make a sketch of the graph that's consistent with all of this information.

*Being able to "sketch" a graph means having some rough picture, either in your head or on a scrap of paper, of what the graph *ought* to look like when drawn more accurately by a computer. Of course we want to use computers to draw graphs of functions, but we also want to be able to recognize when a computer-drawn graph is inaccurate. We do that by developing a sense of what kinds of features the graph of a function can have.*

This function has an odd number of terms, so it's not group-able, and there's no greatest common factor (GCF), so it's a good candidate for using the** rational root theorem** with the set of possible rational roots: {±1, ±2}. If none of those work, **f(x)** has no rational roots (this one does, though).

No GCF or grouping here (odd number of terms), so the rational root theorem is probably the way to go, with possible rational roots of ± {1, 2, 3, 6, 9, 18}. Don't forget to look for multiple roots.

An even number of terms suggests that grouping might be a good strategy. Try it here: 2×3 = 6 and 5×3 = 15.

An even number of terms suggests that grouping might work. In this example, 1×2 = 2 and 3×2 = 6, which is enticing ...

This looks like a good candidate for using the **rational root theorem**: Make a list of possible rational roots, p/q = ±{1, 3, 9}, (always remember that *none* of them may work!) and test them by synthetic substitution (because it's quick).

Don't forget to look for a greatest common factor (GCF). In this case, you can factor out an x (and in so doing discover that x = 0 is a root). An even number of terms suggests the possibility of grouping, too.

In this example, we divide a binomial into a cubic function. It divides evenly, so the binomial is a factor of the function.

In this example, we divide a binomial into a quadratic function. It doesn't go in evenly, so there is a remainder. While this might be seen as a "failure" to find a factor, it is a handy way of rewriting a complicated-looking rational function in a simpler way.

In this example, we use polynomial long division in a slightly different way, to re-express the rational function **f(x) = 1/(x - 2)** as an infinite series of summed terms. You won't likely need this in your early studies of algebra, but it's a neat introduction to the very important field of infinite series that you'll encounter in your studies of calculus.

Time: 3:41

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