Checking remainder through factor
If a polynomial p(x) is divided by x-a, then the remainder is p(a). If p(a)=0, then x-a is a factor of the polynomial.
Practice This ConceptMain explanation
Teacher explanation
This is one of the fastest checks in the chapter. Instead of doing full division every time, we can substitute a into the polynomial. If the result is zero, then x-a is a factor. If the result is not zero, that number is the remainder. This idea links factor theorem and remainder theorem closely.
Example
For p(x)=x^2-4x+3, p(1)=0, so x-1 is a factor.
Simple analogy
Factor x-a means test a.
Common confusion
Students often reverse the factor sign and test the wrong value. For x-a, the test value is a, not -a.
Exam tip
When the divisor is x-a, substitute a directly into p(x).
Answer writing and exam use
1-mark use
Write the exact meaning of checking remainder through factor in one clean line.
2-mark use
Define checking remainder through factor and add one example or condition.
3-mark use
Explain checking remainder through factor, show the method or example, and mention the common mistake.
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