To execute so, you would need to element the denominator come $(x^2+9)(x+3)(x-3)$ and cancel the end $(x-3)$. If i wouldn"t have actually seen it being factored this way, it would"ve to be pretty difficult for me to aspect the denominator in 3 terms. Is there a basic way/method to factor instances like that, that would job-related in most cases?

Thanks in advance!



One of my favored trick as soon as I watch a border for $x o a$ is to make $x=y+a$ and also work the limit for $y o 0$.

You are watching: Factor x^4-81

So, in her case$$fracx^4 - 3x^3x^4-81=frac27 y+27 y^2+9 y^3+y^4108 y+54 y^2+12 y^3+y^4 o frac 27108=frac 14$$


The an initial thing you try to evaluate the limit together $x o 3$ is come boldly shot and plug in $x=3$. You will an alert that both numerator and also denominator evaluate to $0$ there. This implies that performaing polynomial department by $x-3$ will succeed without remainder for both, whereafter you can cancel.


Hopefully you understand that $a^2-b^2=(a+b)(a-b)$

We have the right to use this on the early equation to gain $$x^4-81=(x^2+9)(x^2-9)$$

We can then use the same method on the 2nd bracket to get $$x^2-9=(x+3)(x-3)$$

This gives us our last answer $$x^4-81=(x^2+9)(x+3)(x-3)$$


In this particular problem, as provided by others, the is more than likely simplest to variable using distinction of two squares twice.

More generally, though, if you space taking the limit of a quotient that polynomials together $x o c$ and get a $frac00$ form, then since $c$ is a zero of the top and of the bottom, the factor Theorem states that $x-c$ will certainly be a variable of both the top and bottom. This variable can be taken out, perhaps with man-made division, and also cancelled. (You don"t require to totally factor the numerator and also denominator necessarily, just gain them to the allude of gift able to cancel the $x-c$ factor.)

Write this as $lim_x o 3 fracx^3(x-3)(x-3)(x+3)(x^2+9)=lim_x o 3 fracx^3(x+3)(x^2+9)=frac27108=frac14$

Thanks for contributing response to historicsweetsballroom.comematics Stack Exchange!

Please be certain to answer the question. Provide details and also share your research!

But avoid

Asking because that help, clarification, or responding to various other answers.Making statements based upon opinion; ago them up with recommendations or an individual experience.

Use historicsweetsballroom.comJax to format equations. historicsweetsballroom.comJax reference.

See more: What Kind Of Paint To Use On Styrofoam, What Is The Best Paint To Use On Styrofoam

To find out more, see our tips on writing good answers.

short article Your price Discard

By click “Post your Answer”, friend agree come our regards to service, privacy policy and also cookie plan

Not the prize you're feather for? Browse other questions tagged algebra-precalculus factoring or questioning your very own question.

site architecture / logo © 2021 ridge Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.12.20.41044

your privacy

By click “Accept all cookies”, you agree stack Exchange can store cookies on your an equipment and disclose information in accordance with our Cookie Policy.