Worked example 1
Use the remainder theorem
Find the remainder when f(x) = x³ − 4x + 1 is divided by x − 2.
- For x − 2, substitute x = 2.
- Calculate f(2) = 8 − 8 + 1
Answer: Remainder = 1
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Higher Mathematics
Factor and remainder theorem
Evaluate f(a), identify factors and use remainders to reason about polynomials.
Before you start
- Substitute signed values accurately into powers.
- Know that a factor gives a zero value.
- Keep polynomial working organised term by term.
Explanation
The remainder theorem says that when f(x) is divided by x − a, the remainder is f(a). The factor theorem is the special case where f(a) = 0, so x − a is a factor.
This is a fast way to test roots, find unknown coefficients and begin solving polynomial equations.
Method and rules
- Remainder on division by x − a is f(a).
- If f(a) = 0, then x − a is a factor.
- If x + a is the factor, test f(-a).
Worked examples
Worked example 2
Test a factor
Show that x + 1 is a factor of f(x) = x³ + 2x² − x − 2
- For x + 1, substitute x = −1.
- f(-1) = −1 + 2 + 1 − 2 = 0
So: Since f(-1) = 0, x + 1 is a factor
Watch out
- Testing x = a for the factor x + a
- Confusing a zero remainder with a non-zero remainder.
- Dropping brackets around negative substitutions.
- Stopping after finding a factor when the question asks for roots.
Exam reminder
Write the substitution value clearly. For x − a use f(a); for x + a use f(-a)