Factor a^4+b^4: Simplify \(a^4 + b^4 \)

  • MHB
  • Thread starter Dustinsfl
  • Start date
  • Tags
    Factoring
In summary, the conversation discusses factoring a complex equation with powers higher than two. It is mentioned that while $x^2+y^2$ does not factor over the reals, $x^4+y^4$ does. The conversation also touches on different ways to factor the equation, including using $(a^2+b^2)^2-2a^2b^2$.
  • #1
Dustinsfl
2,281
5
I am trying to write
\[
\frac{a^4+b^4}{a^2+b^2}
\]
with nothing higher than a power of two.

I know \(a^2+b^2 = (a + ib)(a - ib)\) and \(a^4 + b^4 = (a^2 + ib^2)(a^2 - ib^2)\), but I am to take the numerator down in farther in hopes of some cancelling in the denominator.
 
Mathematics news on Phys.org
  • #2
It's a little-known fact that while $x^{2}+y^{2}$ does not factor over the reals, $x^{4}+y^{4}$ does. In fact,
$$x^{4}+y^{4}=(x^{2}+ \sqrt{2} xy+y^{2})(x^{2}- \sqrt{2} xy+y^{2}).$$
 
  • #3
Ackbach said:
It's a little-known fact that while $x^{2}+y^{2}$ does not factor over the reals, $x^{4}+y^{4}$ does. In fact,
$$x^{4}+y^{4}=(x^{2}+ \sqrt{2} xy+y^{2})(x^{2}- \sqrt{2} xy+y^{2}).$$

So there won't be any cancelling. Since it ask for powers less than two, I could use my factoring just as well then?
 
  • #4
dwsmith said:
So there won't be any cancelling. Since it ask for powers less than two, I could use my factoring just as well then?

Sure. If you factor the numerator the way I have described, there won't be any powers written that are higher than $2$.
 
  • #5
Ackbach said:
Sure. If you factor the numerator the way I have described, there won't be any powers written that are higher than $2$.

I could also factor the numerator the way I factored it too. If not, why?
 
  • #6
Perhaps use $(a^2+b^2)^2-2a^2b^2$?
 
  • #7
dwsmith said:
I could also factor the numerator the way I factored it too. If not, why?

Well, that would depend on whether you can factor over the complexes or not. If you can factor over the complexes, then you're fine. Otherwise, if you're going to factor, you'd have to use "my" factorization.
 

FAQ: Factor a^4+b^4: Simplify \(a^4 + b^4 \)

What is the simplest form of the expression a^4+b^4?

The simplest form of the expression a^4+b^4 is (a^2+b^2)(a^2-b^2). This is known as the difference of squares and can be factored further using the difference of squares formula (a^2-b^2)=(a+b)(a-b).

How do you factor a^4+b^4?

To factor a^4+b^4, you can use the difference of squares formula (a^2-b^2)=(a+b)(a-b). This will give you (a^2+b^2)(a^2-b^2). Then, you can factor the second term using the same formula to get the final answer of (a^2+b^2)(a+b)(a-b).

Can you simplify a^4+b^4 further?

No, a^4+b^4 is already in its simplest form. It cannot be factored any further using any known formulas or methods.

What is the difference between factor and simplify?

Factoring is the process of breaking down an expression into smaller, simpler expressions that can be multiplied together to get the original expression. Simplifying is the process of reducing an expression to its most basic form. In the case of a^4+b^4, factoring involves breaking it down into smaller expressions, while simplifying involves reducing it to its simplest form.

Can a^4+b^4 be rewritten as a product of two binomials?

Yes, a^4+b^4 can be rewritten as (a^2+b^2)(a^2-b^2), which is a product of two binomials. This is known as the difference of squares and can be further factored using the same formula to get (a^2+b^2)(a+b)(a-b).

Similar threads

Replies
41
Views
2K
Replies
3
Views
763
Replies
19
Views
2K
Replies
2
Views
1K
Replies
3
Views
809
Replies
2
Views
1K
Back
Top