Problem
ALG-B1-M04-P007 Common Root
#7
★★★☆☆ Level 3 of 5
The polynomials \(x^2+ax+b\) and \(x^2+bx+a\) have a common root, with \(a\ne b\). Prove that this root is \(1\).
Subtract the values of the two polynomials at the common root.
Let the common root be \(r\). Then \(r^2+ar+b=0\) and \(r^2+br+a=0\). Subtracting gives \((a-b)(r-1)=0\). Since \(a\ne b\), \(r=1\).
Important idea: a common root is often revealed by subtraction.