Are These Polynomial Factorizations Correct in Z7?

[ertagon2]

Can someone check if my answers are right and help me with the missing ones.
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[HOI]

Oh, dear. You have left almost all of them blank and those you have tried to answer, the "multiple choice" questions, are mostly wrong.

The first question asks you to find the Greatest Common Divisor of \(\displaystyle f(x)= x^4+ x^3+ x+ 1\) and \(\displaystyle g(x)= x^5- x^3+ x^2- 1\). We need to factor those to see what possible divisors there are. The first thing I notice is that \(\displaystyle f(-1)= 1- 1- 1+ 1= 0\) which means f has x+ 1 as a factor. Dividing f by x+ 1 gives \(\displaystyle x^3+ 1\). But \(\displaystyle (-1)^3+ 1= -1+ 1= 0\) also so there is another factor of x+ 1. Dividing \(\displaystyle x^3+ 1\) by x+ 1 we get \(\displaystyle x^2- x+ 1\). That does not factor (in the real numbers) since the quadratic formula give complex roots. \(\displaystyle f(x)= x^4+ x^3+ x+ 1= (x+ 1)^2(x^2- x+ 1)\).

And I see that \(\displaystyle g(1)= 1- 1+ 1- 1= 0\) so g(x) has x- 1 as a factor. In fact \(\displaystyle g(x)= (x- 1)(x^4+ x^3+ x+ 1)\). And [tex](-1)^4+ (-1)^3+ (-1)+ 1= 1- 1- 1+ 1= 0 so we have an additional factor of x+ 1: \(\displaystyle g(x)= (x- 1)(x+ 1)(x^3+ 1)\). Again \(\displaystyle (-1)^3+ 1= -1+ 1= 0\) so there is another factor of x+ 1: \(\displaystyle g(x)= (x- 1)(x+ 1)^2(X^2- x+ 1)\). The quadratic formula again shows that \(\displaystyle x^2- x+ 1\) does not factor. \(\displaystyle g(x)= x^5- x^3+ x^2- 1= (x- 1)(x+ 1)^2(x^2- x+ 1)\).

So what factors do f(x) and g(x) have in common?

If you expect help with the others, show what you have tried so we will understand what you do know about them!

 

[HOI]

For number 9, "word 5" is "every".

Let P(n) b statement about the natural numbers. If
1. P(1) is true[/b] and
2. P(k) implies P(k+1) for every integer k> 0

then we can conclude that P(n) is true for every $n\in N$.

 

[HOI]

For 10, you have the statement "Every proof by mathematical induction requires at least two base cases to be checked" marked "true". That is incorrect. Proof by induction requires only one "base case" be checked.

And you have the statement "Mathematical induction differs from the kind of induction used in the experimental sciences because it is actually a form of deductive reasoning" marked "false". That is incorrect. "Mathematical Induction" is "deductive reasoning" while the type of "induction" used in the experimental sciences is "inductive reasoning".

 

[HOI]

In 7, $x^3- 1= (x- 1)(x^2- x+ 1)$ and $x^4- x^3+ x^2- 1= (x- 1)(x^3+ x+ 1 )$.
The two polynomials, $x^2- x+ 1$ and $x^3+ x+ 1$ are "irreducible" over the natural numbers. But working in $Z_7$, we need to check their values for x= 0, 1, 2, 3, 4, 5, and 6 "mod 7". $3^2- 3+ 1= 7$ and $5^2- 5+ 1= 25- 5+ 1= 21$, a multiple of 7 so $x^2- x+ 1= (x- 5)(x- 3)$ (mod 7) so we can write $x^3- 1= (x- 1)(x- 3)(x- 5)$ mod 7. No value from 0 to 6 makes $x^3+ x+ 1$ a multiple of 7 so it is irreducible even in $Z_7$.

In $Z_7$ $x^3- 1= (x- 1)(x- 3)(x- 5)$ and $x^4- x^3+ x^2- 1= (x- 1)(x^3+ x+ 1)$. What is the greatest common divisor of those?