BEZOUT's IDENITY solution there just confused on few steps.Help please

[Simkate]

I have a Questions for Bezout's Identity of POLYNOMIALS: I have the soltuions however i am confused about some things in between the answers HELP please...Thank You

The question was Find the (g.c.d) of (( x^5 +1), (x^3 +1)) = ( x+1) ( i found) by the Euclids Algorithim by the following steps:

x^5 +1 = (x^3 +1)(x²) +(x²+1)
x^ 3 +1= (x²+1)(x) + (x+1)
x² + 1 = (x+1)(x+1) + (0)

so G.C.D =(x+1)

What i don't get is how is
x^5 +1 = (x^3 +1)(x²) +(x²+1)^5
= x^5+x² + x² +1
= x^5+ 2x² + 1
= x^5 + 1

How does 2x² = 0? Why?

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My other confusion is : using the Bezout's identity for the above problem

x+1 = (x^3+1)+(x²+1)(x)
= (x^3+1)+((x^5 +1) ( x^3+1)x²)x
= (x^5+1)(x)+(x^3+1)(x^3 +1)

the last step it equals (x^5+1)(x)+(x^3+1)(x^3 +1)

I do not understand how the x^3 in the 2nd step =0 and how ( x^3+1)(x^3 +1) Formed?

 

[lippyka]

The steps of the Euclidean algorithm show that x^5+1 is divisible by x+1, and this is the greatest common divisor of the two polynomials. In terms of Bezout's identity, this means that there exist coefficients a and b such that: a(x^5+1) + b(x^3+1) = x+1 Substituting in the original equations for x^5+1 and x^3+1 gives: ax^5 + a + bx^3 + b = x+1 Multiplying out the brackets on the left side gives: ax^5 + (a+bx²)x + bx + b = x+1 Comparing coefficients gives the following equations: a=1, b=-1 Therefore, the Bezout's identity can be written as: x^5 + 1 - (x^3 +1)(x) = x+1 To answer your questions: 2x² = 0 because b=-1 and so (a+bx²) = 1-x² = 0. The last step is formed by rearranging the equation to put it in the form of a Bezout's identity: x^5 + 1 - (x^3 +1)(x) = x+1 Which can also be written as: (x^5+1)(x)+(x^3+1)(x^3 +1) = x+1