Question

The equation $12x^4-56x^3+89x^2-56x+12=0$

• A. has all roots integral
• B. has all roots irrational
• C. has all roots rational
• D. two roots real, two roots imaginary

Answer 1

The correct option is C

has all roots rational

Given equation

$12x^4-56x^3+89x^2-56x+12=0$
Divide both sides by $x^2$

$12x^2-56x+89-\frac{56}{x}+\frac{12}{x^2}=0$

Rearrange the terms

$12(x^2+\frac{1}{x^2})-56(x+\frac{1}{x})+89=0$ ________ (1)

Let's x + $\frac{1}{x}=y$

$12(y^2-2)-56y+89=012y^2-24-56y+65=0(16y-13)(2y-5)=0y=\frac{13}{6},y=\frac{5}{2}x+\frac{1}{x}=\frac{13}{6},x+\frac{1}{x}=\frac{5}{2}$

$\begin{array}{c}6x^2-13x+6=0\\ x=\frac{2}{3},\frac{3}{2}\end{array}|\begin{array}{c}2x^2-5x+2=0\\ x=2,\frac{1}{2}\end{array}$

Hence roots of the equation are $\frac{2}{3},\frac{3}{2},2,\frac{1}{2}$

So, all the roots are rational number.