WEBVTT
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[MUSIC PLAYING]
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Hello.
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Let us consider
the first exercise
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of the "Roots of polynomials
in practice" step.
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The exercise is asking
to give examples
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of cubic and
quartic polynomials,
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with all the possible
numbers of different roots.
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Of course, there are a
lot of possible examples.
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Now we will see some of them.
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OK.
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First of all, let me divide
the blackboard in two parts
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to consider the case
of cubic polynomials
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and quartic polynomials.
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First of all, remember that
a polynomial of degree n
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has at most n roots.
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Therefore, for a
cubic polynomial,
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we have the following
possibilities.
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The number of roots
could be 0, 1, 2, or 3.
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Quartic polynomials,
the roots can
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be 0, 1, 2, 3 and 4 at most.
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Then remember that over real
numbers, all the polynomials
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of odd degree, all they
have at least one root.
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Therefore, there are no cubic
polynomials with 0 roots.
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Now let us give an example of a
cubic polynomial with just one
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root.
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OK?
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For example, x to the cube.
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This is a polynomial
which has only 0 as root.
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In this case, you see
0 is what we usually
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say a root of multiplicity 3.
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But for example, you
can also consider
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the following polynomial,
x times x squared plus 1.
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And you see, x squared
plus 1 is always
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greater or equal than 1 for each
real number I substitute to x.
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Therefore, this factor has
no roots, no real roots.
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Hence this cubic polynomial
has only one root,
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x equals 0, now with
multiplicity equal 1.
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And now let us consider
the case of two roots.
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OK?
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For example, we can consider
x times x minus 1 squared.
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This polynomial has exactly
two different roots, x equal 0
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and x equal 1.
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x equal 1 is a root of
multiplicity 2, x equal 0
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is a root of multiplicity 1.
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And finally, a cubic with
three different roots
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could be x times x
minus 1 times x minus 2.
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And now let us consider
the quartic polynomials.
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Now for a quartic
polynomial, it's
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possible to have
no roots at all.
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Consider for example x
to the fourth plus 1.
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This is always greater or equal
than 1 for each real number we
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decide to substitute to x.
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Therefore, this
polynomial has no roots.
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With one root, we can
consider x to the fourth.
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Again, 0 is the only
root of this polynomial.
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And 0 is the root
of multiplicity 4.
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Now, with two different
roots, then we
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can consider x squared
times x minus 1 squared.
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Then in this situation, we
have only two different roots,
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x equals 0 and x equal 1,
both with multiplicity 2.
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Or otherwise, we
can also consider
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x times x minus 1
times x squared plus 1.
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Again, this factor has no roots.
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And therefore, this
quartic polynomial
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has exactly two roots, each
one a multiplicity of 1.
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Now a quartic with three roots.
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Then we can consider
x squared times x
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minus 1 times x minus 2.
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OK?
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This is a polynomial with
exactly three roots--
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0 of multiplicity 2,
1 of multiplicity 1,
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and 2 of multiplicity 1.
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And finally, let us see
a quartic polynomial
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with four different roots.
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x times x minus 1 times x
minus 2 times x minus 3.
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This is an example
of a polynomial
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with four different roots.
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The roots are 0, 1, 2, and
3, all of multiplicity 1.
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