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 "Polynomial
division in practice" step.
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We have to divide this
polynomial by this one.
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Then we have to
write the first one--
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x to the fourth minus x to the
third plus x squared minus 3
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times x minus 1.
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Then we consider this
line and this other line.
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And here, now we
write the polynomial x
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squared plus 1, for which we
want to divide this polynomial.
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OK, which is the idea now?
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I have to write
down here something
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that multiplied by
x squared gives me
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the term of maximum
degree in this expression,
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which is this one--
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x to the fourth.
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OK, then clearly, if I
write here x squared,
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we have that the x
squared times x squared is
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exactly equal x to the fourth.
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OK, and now what do we do?
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Then we make this multiplication
and this multiplication,
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and we write under this
polynomial what we get.
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We get x squared
times x squared,
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which is x to the fourth
plus 1 times x squared,
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which is plus x squared.
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You see, I have
written the new terms
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under the previous terms
of the same degree.
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And now I have to
do the subtraction
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between this polynomial
and this polynomial.
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And what do we get?
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x to the fourth minus
x to the fourth is 0.
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Minus x to the 3 minus 0
is minus x to the three.
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x squared minus x squared is 0.
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And then we have
minus 3x minus 1.
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And now I repeat exactly
the same algorithm.
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Now I am asking myself: for what
I have to multiply this term
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to get the term of maximum
degree now in this new line.
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You see?
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I have to multiply x
squared times minus x
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to get minus x cubed.
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OK, let us go on.
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x squared times minus
x is minus x cubed.
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And then I have plus 1 times
minus x, which is minus x.
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And again, I consider the
subtraction of these two
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polynomials, and I have 0 here.
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And here, I have
minus 3x and minus
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minus x, which is
minus 2x minus 1.
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And now you see I have
reached a polynomial which
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is of degree less than 2--
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less than the degree of
the polynomial for which
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we are making our division.
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Therefore, we can
stop, and we can
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conclude that x to the fourth
minus x to the 3 plus x squared
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minus 3 times x minus 1 is
equal to x squared plus 1 times
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x squared minus x
plus the remainder,
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which is minus 2x minus 1.
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OK.
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Thank you.
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Hello.
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Let us consider
the second exercise
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of the "Polynomial
division in practice" step.
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It's another division
between two polynomials.
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Then let us apply the
same algorithm as before--
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as exercise 1.
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Then we write x cubed
minus 8 on the side.
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Then we consider this
line, another line.
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And here, I write x minus 2.
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OK, I have now to find
something that multiplied
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by x gives me x cubed.
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This is x squared.
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And now I make
the multiplication
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of this polynomial
by this guy, and I
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get x times x squared, which
is equal to x cubed and minus 2
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times x squared, which
is minus 2 x squared.
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And now I take the subtraction
of these two polynomials.
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The subtraction
between two polynomials
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is done considering
the subtraction
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of terms with the same degree.
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And then I have x cubed
minus x cubed, which is 0.
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Then here, there is
no terms of degree 2
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minus minus 2 x squared,
which is 2 x squared.
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And then I have minus 8.
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Here, there are no
terms of degree 0.
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And then I get just minus 8.
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OK, and now I repeat
the same argument.
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For what I have to multiply
x to get 2 x squared?
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Of course, I have
to multiply x by 2x.
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And in such a way, I get x
times 2x, which is 2 x squared,
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and minus 2 times 2x,
which is minus 4x.
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And I again consider
the subtraction
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of these two polynomials, and
I immediately get 4x minus 8.
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Attention: the degree
of this polynomial
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is not strictly
less than the degree
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of this polynomial, the
polynomial for which I
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am making a division.
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Therefore, I cannot stop here.
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I have to continue.
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And then now, I ask
myself, for what
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I have to multiply x to get 4x?
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Of course, by 4--
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and I get x times 4, which
is 4x minus 2 times 4, which
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is minus 8.
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And when I consider here
the subtraction of these two
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polynomials,
immediately I get 0.
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What means?
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This means that the remainder
of this division is equal to 0.
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Then we have that x cubed minus
8 is equal to x minus 2 times
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x squared plus 2x plus 4 plus 0.
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Therefore, I can conclude
that this equality is true.
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Ciao.
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