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
Higher Mathematics
Higher Maths Factor and remainder theorem
Practise factor and remainder theorem for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Evaluate f(a), identify linear factors and find remainders of polynomial division.
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)
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.