Math Is Fun Forum

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Show Binomial Is Prime

Show that x^2 + 4 is prime.

Problem 133; page 58.

Let me see.

I say x^2 + 4 is not a prime polynomial. A prime polynomial is one that cannot be factored into the product of two non-constant polynomials with number coefficients.

In this case, x^2 + 4 can be factored as (x+2i)(x-2i), where i is the imaginary unit defined by i^2 = -1. Since both (x+2i) and (x-2i) are non-constant polynomials with number coefficients, I think I have shown that x^2 + 4 is not a prime polynomial.

You say?

Last edited by mathxyz (2024-05-20 07:06:47)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Show Binomial Is Prime

Jumping into complex numbers is brilliant and a correct factorisation But the question says show prime ?  I think you weren't supposed to  go there but stick to real numbers. 

Have a look at P53 example 13

Proir to learning about complex numbers my teachers used to say 'cannot factorise' for questions like this.

Bob

x² + 4

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Show Binomial Is Prime

Jumping into complex numbers is brilliant and a correct factorisation But the question says show prime ?  I think you weren't supposed to  go there but stick to real numbers. 

Have a look at P53 example 13

Proir to learning about complex numbers my teachers used to say 'cannot factorise' for questions like this.

Bob

x² + 4

Ok. Thanks. I didn't know that Sullivan has an example in his book. I will tale a look later. I am not home.

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Show Binomial Is Prime

Jumping into complex numbers is brilliant and a correct factorisation But the question says show prime ?  I think you weren't supposed to  go there but stick to real numbers. 

Have a look at P53 example 13

Proir to learning about complex numbers my teachers used to say 'cannot factorise' for questions like this.

Bob

x² + 4

Ok. Here I go.

(-2) + (-2) = -4

2 + 2 = 4

(-1) + (-4) = -5

1 + 4 = 4

Notice that none of the sums are equal to 0.

The coefficient of the middle term for x^2 + 0x + 4 is also 0.

I conclude that x^2 + 4 is prime.

You say?

Last edited by mathxyz (2024-05-23 20:35:58)

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Show Binomial Is Prime

Jumping into complex numbers is brilliant and a correct factorisation But the question says show prime ?  I think you weren't supposed to  go there but stick to real numbers. 

Have a look at P53 example 13

Proir to learning about complex numbers my teachers used to say 'cannot factorise' for questions like this.

Bob

x² + 4

Bob,

Sullivan gives examples in his textbook lecture notes and then highlights a few selected problems to work out. I will only work out the suggested problems per section. Seriously, if I try to answer every single problem in both textbooks, I will not begin calculus until my retirement age (whenever that is). So, this means less problems posted but more learning taking place.

I will also select 1 or 2 interesting questions (without Sullivan samples) per section per textbook as a personal challenge. I think this will be more than enough math for what I should know. Of course, if I know how to solve the highlighted Sullivan questions, then there's no need to post here.

The goal is to increase my skills without going through every single problem in the textbooks. Honestly, who has time for that? Supposedly, I start my new overnight job tonight, which means even less time for math, guitar and anything else I enjoy doing.