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Polynomial division in practice

Solving exercises about the euclidian division algorithm for polynomials
10.8
Hello. Let us consider the first exercise of the “Polynomial division in practice” step. We have to divide this polynomial by this one. Then we have to write the first one– x to the fourth minus x to the third plus x squared minus 3 times x minus 1. Then we consider this line and this other line. And here, now we write the polynomial x squared plus 1, for which we want to divide this polynomial. OK, which is the idea now? I have to write down here something that multiplied by x squared gives me the term of maximum degree in this expression, which is this one– x to the fourth.
78.6
OK, then clearly, if I write here x squared, we have that the x squared times x squared is exactly equal x to the fourth. OK, and now what do we do? Then we make this multiplication and this multiplication, and we write under this polynomial what we get. We get x squared times x squared, which is x to the fourth plus 1 times x squared, which is plus x squared. You see, I have written the new terms under the previous terms of the same degree. And now I have to do the subtraction between this polynomial and this polynomial. And what do we get? x to the fourth minus x to the fourth is 0.
139.7
Minus x to the 3 minus 0 is minus x to the three. x squared minus x squared is 0. And then we have minus 3x minus 1. And now I repeat exactly the same algorithm.
162.1
Now I am asking myself: for what I have to multiply this term to get the term of maximum degree now in this new line. You see? I have to multiply x squared times minus x to get minus x cubed.
183.7
OK, let us go on. x squared times minus x is minus x cubed. And then I have plus 1 times minus x, which is minus x. And again, I consider the subtraction of these two polynomials, and I have 0 here. And here, I have minus 3x and minus minus x, which is minus 2x minus 1. And now you see I have reached a polynomial which is of degree less than 2– less than the degree of the polynomial for which we are making our division.
239.7
Therefore, we can stop, and we can conclude that x to the fourth minus x to the 3 plus x squared minus 3 times x minus 1 is equal to x squared plus 1 times x squared minus x plus the remainder, which is minus 2x minus 1.
274.9
OK. Thank you.
279.1
Hello. Let us consider the second exercise of the “Polynomial division in practice” step. It’s another division between two polynomials. Then let us apply the same algorithm as before– as exercise 1. Then we write x cubed minus 8 on the side. Then we consider this line, another line. And here, I write x minus 2. OK, I have now to find something that multiplied by x gives me x cubed. This is x squared. And now I make the multiplication of this polynomial by this guy, and I get x times x squared, which is equal to x cubed and minus 2 times x squared, which is minus 2 x squared. And now I take the subtraction of these two polynomials.
348.5
The subtraction between two polynomials is done considering the subtraction of terms with the same degree. And then I have x cubed minus x cubed, which is 0. Then here, there is no terms of degree 2 minus minus 2 x squared, which is 2 x squared. And then I have minus 8. Here, there are no terms of degree 0. And then I get just minus 8. OK, and now I repeat the same argument. For what I have to multiply x to get 2 x squared? Of course, I have to multiply x by 2x.
400.5
And in such a way, I get x times 2x, which is 2 x squared, and minus 2 times 2x, which is minus 4x. And I again consider the subtraction of these two polynomials, and I immediately get 4x minus 8.
427.1
Attention: the degree of this polynomial is not strictly less than the degree of this polynomial, the polynomial for which I am making a division. Therefore, I cannot stop here. I have to continue. And then now, I ask myself, for what I have to multiply x to get 4x? Of course, by 4– and I get x times 4, which is 4x minus 2 times 4, which is minus 8. And when I consider here the subtraction of these two polynomials, immediately I get 0. What means? This means that the remainder of this division is equal to 0. Then we have that x cubed minus 8 is equal to x minus 2 times x squared plus 2x plus 4 plus 0.
497.2
Therefore, I can conclude that this equality is true.
504.1
Ciao.
The following exercise is solved in this step.
We invite you to try to solve it before watching the video.
In any case, you will find below a PDF file with the solution.

Exercise 1.

Divide the polynomial \(x^4-x^3+x^2-3x-1\) by \(x^2+1\).

Exercise 2.

Divide the polynomial \((x^3-8)\) by \((x-2)\).

Exercise 3. [Solved only in the PDF file]

Make the following divisions between polynomials:
i) \((2x^3-3x^2+3x+2) : (2x-1)\);
ii) \((x^4+2x^3-2x^2-5):(x+x^2-4)\).
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Precalculus: the Mathematics of Numbers, Functions and Equations

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