<s>[INST] How to factorise $x^4 - 3x^3 + 2$, so as to compute the limit of a quotient? Question:

Find the limit: $$\lim_{x \to 1}\frac{x^4 - 3x^3 + 2}{x^3 -5x^2+3x+1}$$

The denominator can be simplified to: $$(x-1)(x^2+x)$$
However, I am unable to factor the numerator in a proper manner (so that $(x-1)$ will cancel out)
I know upon graphing that the limit is $5\over4$. What should I do here?
Note: To be done without the use of L'Hospital Rule
 [/INST] You may use Horner here. Since $x=1$ is a root of the nominator (you can check that easily) then you can deduce that down by applying a long division.
Hence your limit is deduced down to:
$$\lim_{x\rightarrow 1}\frac{x^4-3x^3+2}{x^3-5x^2+3x+1}=\lim_{x\rightarrow 1}\frac{(x-1)\left ( x^3-2x^2-2x-2 \right )}{\left ( x-1 \right )\left ( x^2-4x-1 \right )}=\frac{1-2-2-2}{1-4-1}=5/4$$
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