Skip to 0 minutes and 12 secondsHello. Let us consider the first exercise of the "Roots of polynomials in practice" step. The exercise is asking to give examples of cubic and quartic polynomials, with all the possible numbers of different roots. Of course, there are a lot of possible examples. Now we will see some of them. OK. First of all, let me divide the blackboard in two parts to consider the case of cubic polynomials and quartic polynomials. First of all, remember that a polynomial of degree n has at most n roots. Therefore, for a cubic polynomial, we have the following possibilities. The number of roots could be 0, 1, 2, or 3. Quartic polynomials, the roots can be 0, 1, 2, 3 and 4 at most.

Skip to 1 minute and 40 secondsThen remember that over real numbers, all the polynomials of odd degree, all they have at least one root. Therefore, there are no cubic polynomials with 0 roots. Now let us give an example of a cubic polynomial with just one root. OK? For example, x to the cube. This is a polynomial which has only 0 as root. In this case, you see 0 is what we usually say a root of multiplicity 3. But for example, you can also consider the following polynomial, x times x squared plus 1. And you see, x squared plus 1 is always greater or equal than 1 for each real number I substitute to x. Therefore, this factor has no roots, no real roots.

Skip to 2 minutes and 47 secondsHence this cubic polynomial has only one root, x equals 0, now with multiplicity equal 1. And now let us consider the case of two roots. OK? For example, we can consider x times x minus 1 squared. This polynomial has exactly two different roots, x equal 0 and x equal 1. x equal 1 is a root of multiplicity 2, x equal 0 is a root of multiplicity 1. And finally, a cubic with three different roots could be x times x minus 1 times x minus 2. And now let us consider the quartic polynomials. Now for a quartic polynomial, it's possible to have no roots at all. Consider for example x to the fourth plus 1.

Skip to 3 minutes and 57 secondsThis is always greater or equal than 1 for each real number we decide to substitute to x. Therefore, this polynomial has no roots. With one root, we can consider x to the fourth. Again, 0 is the only root of this polynomial. And 0 is the root of multiplicity 4. Now, with two different roots, then we can consider x squared times x minus 1 squared. Then in this situation, we have only two different roots, x equals 0 and x equal 1, both with multiplicity 2. Or otherwise, we can also consider x times x minus 1 times x squared plus 1. Again, this factor has no roots. And therefore, this quartic polynomial has exactly two roots, each one a multiplicity of 1.

Skip to 5 minutes and 5 secondsNow a quartic with three roots. Then we can consider x squared times x minus 1 times x minus 2. OK? This is a polynomial with exactly three roots-- 0 of multiplicity 2, 1 of multiplicity 1, and 2 of multiplicity 1. And finally, let us see a quartic polynomial with four different roots. x times x minus 1 times x minus 2 times x minus 3. This is an example of a polynomial with four different roots. The roots are 0, 1, 2, and 3, all of multiplicity 1.

Roots of polynomials in practice

The following exercise is solved in this step.

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Exercise 1.

Give examples of cubic and quartic polynomials with all the possible numbers of different roots.

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Precalculus: the Mathematics of Numbers, Functions and Equations

University of Padova

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