# Interesting math problem involving a weirdly-defined polynomial function **URL:** \ **Category:** Factual Questions\ **Created:** [February 20, 2006, 3:23am UTC](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283 "2006-02-20T03:23:26Z")\ **Posts on this page:** 9\ **Page:** 2
**Author:** ![pikolo77](https://avatars.discourse-cdn.com/v4/letter/p/43a26b/32.png) [@pikolo77](https://boards.straightdope.com/u/pikolo77)\ **Post date:** [February 21, 2006, 6:45pm UTC](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283/21 "2006-02-21T18:45:57Z")
Yes, yes Mentock. I will include the constant because you will it to be: ) Fresh stumper… Can anyone factor x^4 + 2x^2 + 1. This is a wee bit simpler than the one we just worked on. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 21, 2006, 7:11pm UTC](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283/22 "2006-02-21T19:11:29Z")
> [@pikolo77](#): > > Can anyone factor x^4 + 2x^2 + 1. This is a wee bit simpler than the one we just worked on. Over what field? ---
**Author:** ![pikolo77](https://avatars.discourse-cdn.com/v4/letter/p/43a26b/32.png) [@pikolo77](https://boards.straightdope.com/u/pikolo77)\ **Post date:** [February 21, 2006, 7:53pm UTC](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283/23 "2006-02-21T19:53:27Z")
Since this is not an equation (no equals sign) that’s not an issue. I suppose it’s obvious that this cannot have an real zeros. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [February 21, 2006, 7:59pm UTC](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283/24 "2006-02-21T19:59:51Z")
> [@pikolo77](#): > > Since this is not an equation (no equals sign) that’s not an issue. I suppose it’s obvious that this cannot have an real zeros. The field still matters. The polynomial has 0 roots in **R** , 4 in **C** , and 12 in **H** , and 28 in **O**. So which is it? ---
**Author:** ![pikolo77](https://avatars.discourse-cdn.com/v4/letter/p/43a26b/32.png) [@pikolo77](https://boards.straightdope.com/u/pikolo77)\ **Post date:** [February 21, 2006, 8:17pm UTC](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283/25 "2006-02-21T20:17:16Z")
Let’s keep it simple. Keep it in real numbers. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [February 22, 2006, 1:44am UTC](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283/26 "2006-02-22T01:44:22Z")
It may not have any real roots but it can be factored, into (x^2+1)^2. My 7th graders warm up with that one. But what’s O, the octonions? ---
**Author:** ![Mayo\_Speaks](https://avatars.discourse-cdn.com/v4/letter/m/c57346/32.png) [@Mayo\_Speaks](https://boards.straightdope.com/u/Mayo_Speaks)\ **Post date:** [February 22, 2006, 4:11am UTC](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283/27 "2006-02-22T04:11:50Z")
> [@RM Mentock](#): > > Substitute X=-7 and X=-1 in the equation (X+7)\*P(2X) = 8X \* P(X+1), and use the previously known information that P(0)=0 You can’t do this. For X = -7, this would result in division by zero in the original statement [56/( X+7 )]. For X = -1, P( X+1 ) = 0, which is also a divide by zero error [P( 2X )/P( X+1 )]. Even though the rewritten form doesn’t have a dvision by zero issue, the definitional statement is undefined, and therfore you can’t use it. Sorry ☹ ---
**Author:** ![pikolo77](https://avatars.discourse-cdn.com/v4/letter/p/43a26b/32.png) [@pikolo77](https://boards.straightdope.com/u/pikolo77)\ **Post date:** [February 22, 2006, 3:58pm UTC](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283/28 "2006-02-22T15:58:58Z")
> [@RM Mentock](#): > > It may not have any real roots but it can be factored, into (x^2+1)^2. My 7th graders warm up with that one. > > But what’s O, the octonions? Yeah, it’s an intermediate algebra question for sure. A lot of people don’t see the substitution of x^4 = y^2 which then it turns into a standard y^2 + 2y + 1 expression that’s instantly factorable. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [February 22, 2006, 6:00pm UTC](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283/29 "2006-02-22T18:00:53Z")
> [@This space intentionally blank](#): > > You can’t do this. > > For X = -7, this would result in division by zero in the original statement [56/( X+7 )]. > > For X = -1, P( X+1 ) = 0, which is also a divide by zero error [P( 2X )/P( X+1 )]. > > Even though the rewritten form doesn’t have a dvision by zero issue, the definitional statement is undefined, and therfore you can’t use it. > > Sorry We didn’t substitute into the original statement, only in the rewritten form. Since we know P is a polynomial, the transformation from the original to the rewritten form is allowed, and the rewritten form is a true statement for all x. [Previous page](https://boards.straightdope.com/t/interesting-math-problem-involving-a-weirdly-defined-polynomial-function/345283.md?page=1)