Problem
ALG-B1-M02-P008 A Parameter and a Linear Factor
#8
★★☆☆☆ Level 2 of 5
Find \(k\) if \(x^3+kx^2-5x-2k\) is divisible by \(x-2\), and factor the polynomial for this \(k\).
Substitute \(x=2\).
Let \(P(x)=x^3+kx^2-5x-2k\). The condition is equivalent to \(P(2)=0\).
\(P(2)=8+4k-10-2k=2k-2\), hence \(k=1\).
Then \(P(x)=x^3+x^2-5x-2=(x-2)(x^2+3x+1)\).
This prepares for the later full module on polynomials.