Factors of the Polynomial of Degree 4

[Aaron Wang]

Why does the polynomial of degree 4 have the following factors?

$$a+b-c$$ $$a-b+c$$ $$-a+b+c$$ $$a+b+c$$

[debbie]

@aaron-wang said in Factors of the Polynomial of Degree 4:

a+b+c

Oh, wow, this is a nice question! (By the way, only the first three rows are factors of the $\text{area}^2$ formula.... not the $a + b + c.$ )

The fourth-degree polynomial that Prof. Loh is speaking about is the $\text{(area)}^2$ formula. He writes an ugly-looking expression down. But he's not looking at this ugly formula when he realizes that $a + b - c, a - b + c,$ and $-a + b + c$ are factors. He's just thinking about what the area of a triangle would be like if two of the sides added together were to equal the third side. (No algebra involved!! )

 

You see, if $a + b - c = 0,$ then you can't make a triangle. You just get two straight lines that line up along the third line. And we call a "root" of a polynomial another expression that makes the polynomial $0.$

Like for example, consider $x^2 + 5x + 6 = 0.$ When $x = 3,$ then $x^2 + 5x + 6 = 0,$ so $\textcolor{red}{x -3}$ is a root of that polynomial. You can even factor the polynomial out as $(\textcolor{red}{x-3})(x-2) = 0.$

So we know that the area formula looks something like:

$\text{ Area}^2 = (a + b -c)(a - b + c)(-a + b + c) \times \text{(some other stuff....)}.$

Because when one of those terms (factors) equals $0,$ then the area equals $0.$

I hope this helped. Please let me know if there's something I can clarify for you! Thanks again for asking a great question!

[Aaron Wang]

@debbie Ah, I see! Thank you for clarifying this, it definitely helped.

[debbie]

@aaron-wang That's great!

Have a !

[The Blade Dancer]

Can I have a too

[debbie]

@the-blade-dancer for you (munch, munch)

[Aaron Wang]

Hamburgers are cool