If an integer p is a root of f(x)=0 with f(x)=px2−3x−p−3, then the gradient of the tangent line to the curve y=f(x) at the point with abscissa x=p is....
Explanation
The integer p is a root of f(x)=0. We can conclude that x=p so f(p)=0
f(x)=px2−3x−p−3
f(p)=0
p⋅p2−3p−p−3=0
p3−3p−p−3=0
p3−4p−3=0
(p+1)(p2−p−3)=0
p=−1∪p2−p−3=0
The factorization above is obtained using Horner's method.
Since the form p2−p−3=0 cannot be factored further, we only obtain p=−1. Thus the function becomes
