Identities, the remainder theorem, and the factor theorem
Distinguish identities from equations to solve and connect remainder and factor conditions.
Meaning
An identity holds for all allowed values. Dividing P(x) by x−a leaves remainder P(a); if P(a)=0, x−a is a factor. Write polynomial division as P(x)=(x−a)Q(x)+r. Substituting x=a makes the product zero, leaving P(a)=r. If that remainder is zero, P(x) is a product with x−a, so x−a is a factor.
Example
Practice for this concept
Scope and references
Topics
Equations true throughout a domain, comparing coefficients, polynomial remainders and factors
Learning goal
Distinguish identities from equations to solve and connect remainder and factor conditions.
Helpful background
Explore polynomial operations and what it means to hold for all allowed values.
Identities, remainder and factor theorems
These ordinary choice questions provide limited practice in this concept. They do not establish independent solution, full-concept performance, or mastery.