Question

Factorise : a² + 1/a² + 2 – 5a – 5/a.

Answer 1

Identity $(a+\frac{1}{a} )^2=a^2+\frac{1}{a^2}+2$

Thus, $(a+\frac{1}{a} )^2-2+2-5(a+\frac{1}{a} )$.

$(a+\frac{1}{a} )^2-5(a+\frac{1}{a} )$

∴$(a+\frac{1}{a} )(a-5+\frac{1}{a} )$

(similar to the properties of reciprocal polynomial)

ex) $ax^4+bx^3+cx^2+bx+a=0$

$x^2 \{a(x^2+\frac{1}{x^2} )+b(x+1)+c \}$

$x^2 \{a(x+\frac{1}{x} )^2+b(x+\frac{1}{x} )+c-2 \}$ ←←like this