Video Examples

Finding roots & sketching the graphs of polynomial functions

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

  1. Find the roots, if possible
    1. Greatest common factor (GCF), grouping, familiar pattern (quadratic, sum/difference of perfect cubes), rational root theorem
  2. Find the y-intercept, f(0), and any other easy-to-find points (usually just f(0) is necessary)
  3. Determine the end behavior
  4. Remember that all polynomial functions have smooth, continuous graphs
  5. Make a sketch of the graph that's consistent with all of this information.

Examples 7-9 are polynomial long division examples

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.

Example 1:   f(x) = x4 + x3 - 3x2 - x + 2

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).


Example 2:   f(x) = x4 - 3x3 - 7x2 + 27x - 18

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.


Example 3:   f(x) = 2x3 - 6x2 - 5x + 15

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


Example 4:   f(x) = 3x3 - x2 - 6x + 2

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


Example 5:   f(x) = -x5 + 3x4 + 6x3 - 26x2 + 27x - 9

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).


Example 6:   f(x) = x4 - 2x3 - x2 + 2x

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.


Example 7:   Polynomial division (1 of 3)

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


Example 8:   Polynomial division (2 of 3)

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.


Example 9:   Polynomial division (3 of 3)

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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