We learned that a Quadratic Function is a special type of polynomial with degree 2; these have either a cup-up or cup-down shape, depending on whether the leading term (one with the biggest exponent) is positive or negative, respectively. There are certain rules for sketching polynomial functions, like we had for graphing rational functions. Again, the degree of a polynomial is the highest exponent if you look at all the terms (you may have to add exponents, if you have a factored form). The leading coefficient of the polynomial is the number before the variable that has the highest exponent (the highest degree).

So for, the degree is 4, and the leading coefficient is 5; for, the degree is 7 (add exponents since the polynomial isn’t multiplied out and don’t forget the x to the first power), and the leading coefficient is –10 (you can tell by the –5 in front and the 2x in the factor with the highest exponent).

Each factor in a polynomial has what we call a multiplicity, which just means how many times it’s multiplied by itself in the polynomial (its exponent). Remember that x – 4 is a factor, while 4 is a root (zero, solution, x-intercept, or value). Now we can use the multiplicity of each factor to know what happens to the graph for that root – it tells us the shape of the graph at that root.

Also note that you won’t be able to determine how low and high the curves are when you sketch the graph; you’ll just want to get the basic shape.

Here are a few problems where we use the Conjugate Zeroes Theorem and Complex Conjugate Zeroes Theorem (also called Conjugate Root Theorem or Conjugate Pair Theorem), which states that if is a root, then so is . The complex form of this theorem states that if is a root, then so is .

Also remember that when we factor to solve quadratics or any polynomials, we can never just divide by factors (with variables) on both sides to get rid of them. If we do this, we may be missing solutions!

Many times we’re given a polynomial in Standard Form, and we need to find the zeros or roots.

For higher level polynomials, the factoring can be a bit trickier, but it can be sort of fun — like a puzzle! Remember that if we divide a polynomial by “x – c” and get a remainder of 0, then “x – c” is a factor of the polynomial and “c” is a root.

When we want to factor and get the roots of a higher degree polynomial using synthetic division, it can be difficult to know where to start! In the examples so far, we’ve had a root to start, and then gone from there. For a polynomial function with integers as coefficients (no fractions or decimals), if p = the factors of the constant (in our case, d), and q = the factors of the highest degree coefficient (in our case, a), then the possible rational zeros or roots are where p are all the factors of d above, and q are all the factors of a above. Remember that factors are numbers that divide perfectly into the larger number; for example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

The rational root test help us find initial roots to test with synthetic division, or even by evaluating the polynomial to see if we get 0. However, it doesn’t make a lot of sense to use this test unless there are just a few to try, like in the first case above. Now let’s try to find roots of polynomial functions without having a first root to try. Also remember that you may end up with imaginary numbers as roots, like we did with quadratics.

Take out any Greatest Common Factors (GCFs) of the polynomial, and you’ll have to set those to 0 too, to get any extra roots. For example, if you take an x out, you’ll add a root of “0”. If you have access to a graphing calculator, graph the function and determine if there are any rational zeros with which you can use synthetic division. If you don’t have a calculator, guess a possible rational zero using the method above. There are a couple of theorems that you’ll learn about that will help you evaluate polynomials (for a given x, find the y) and also be able to quickly tell if a given number is a root.

There’s another really neat trick out there that you may not talk about in High School, but it’s good to talk about and pretty easy to understand. Yes, and it was named after a French guy! The DesCartes’ Rule of Signs will tell you the number of positive and negative real roots of a polynomial by looking at the sign changes of the terms of that polynomial.

We talked a little bit about the Complex Conjugate Zeros Theorem here when we talked about all the steps required to find all the factors and roots of a polynomial.

Here’s a type of problem that you might see that requires using synthetic division using complex roots. The problem is based on the Conjugate Zeros Theorem. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph. The leading coefficient is significant compared to the other coefficients in the function for the very large or very small numbers. To predict the end-behavior of a polynomial function, first check whether the function is odd-degree or even-degree function and whether the leading coefficient is positive or negative.

Notice that the negative part of the graph is more of a “cup down” and the positive is more of a “cup up”.

If there is no exponent for that factor, the multiplicity is 1 (which is actually its exponent!) And remember that if you sum up all the multiplicities of the polynomial, you will get the degree! So, the sign of the leading coefficient is sufficient to predict the end behavior of the function.

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