Finding HCF of two algebraic expressions

Question

I was working on a problem which required finding Highest Common Factor of following two expressions: $$x^5-13x^4+67x^3-171x^2+216x-108$$ and, $$5x^4-52x^3+201x^2-342x+216$$ Please help me by writing complete solution as I am continuously encountering this problem.

Answer 1

$$x^5-13x^4+67x^3-171x^2+216x-108=(x-2)^2(x-3)^3$$ and $$5x^4-52x^3+201x^2-342x+216=(x-3)^2(x-2)(5x-12).$$ Thus, we got the answer: $$(x-3)^2(x-2).$$

Since $108=3^32^2$ we can try to check firstly that $2$ or $3$ are roots of the polynomial: $$x^5-13x^4+67x^3-171x^2+216x-108$$ and we see that they are indeed roots, which says that we got factors $x-2$ and $x-3$

and since $$(x-2)(x-3)=x^2-5x+6,$$ we obtain: $$x^5-13x^4+67x^3-171x^2+216x-108=$$ $$=x^5-5x^4+6x^3-8x^4+40x^3-48x^2+21x^3-105x^2+126x-18x^2+90x-108=$$ $$=(x^2-5x+6)(x^3-8x^2+21x-18)=$$ $$=(x^2-5x+6)(x^3-5x^2+6x-3x^2+15x-18)=$$ $$=(x^2-5x+6)^2(x-3)=(x-3)^3(x-2)^2.$$ By the same way we can get the factorization of the second polynomial.

Answer 2

Given

$A(x)=x^5-13 x^4+67 x^3-171 x^2+216 x-108$

$B(x)=5 x^4-52 x^3+201 x^2-342 x+216$

You can use the Euclidean algorithm

Divide $A(x)$ by $B(x)$ and get

quotient $Q(x)=\frac{x}{5}-\frac{13}{25}$ and

remainder $R(x)=-\frac{6 x^3}{25}+\frac{48 x^2}{25}-\frac{126 x}{25}+\frac{108}{25}$

$\text{GCD}(A(x),B(x))=\text{GCD}(B(x),R(x))$

Divide now $B(x)$ by $R(x)$ and get quotient

$Q'(x)=50-\frac{125 x}{6}$ and remainder $R'(x)=0$

then we can stop and

as $$R(x)=-\frac{6 x^3}{25}+\frac{48 x^2}{25}-\frac{126 x}{25}+\frac{108}{25}=-\frac{6}{25} \left(x^3-8 x^2+21 x-18\right)$$ then $\text{GCD}(A(x),B(x))=x^3-8 x^2+21 x-18$