LEARNING OBJECTIVES
What you will be able to do
- apply the remainder theorem
- use the factor theorem in both directions
- divide a polynomial by a linear expression
- factorise and solve cubic equations
AT A GLANCE
INTRODUCTION · THE BIG IDEA
Use substitution and division to turn a high-degree expression into solvable factors.
Polynomial questions often hide one accessible linear factor. A quick substitution can test a proposed factor or turn information about a remainder into an equation for an unknown coefficient.
Once a cubic is divided by one linear factor, the remaining quadratic can be handled with familiar methods.
SECTION 01
Remainder and factor theorems
Dividing f(x) by x − a leaves remainder f(a). For a divisor such as 2x + 3, use its zero x = −3/2 in the same way.
Determine an unknown coefficient
- Let f(x) = x³ + kx² − 5x + 6 and suppose x − 2 is a factor.
- Use f(2) = 0.
- Compute 8 + 4k − 10 + 6 = 0.
- Solve 4k + 4 = 0.
Answer: k = −1.
SECTION 02
Polynomial division
Long division matches leading terms, multiplies back and subtracts repeatedly. Include zero placeholders for missing powers so columns remain aligned.
DETAILED EXPLANATION
- The quotient degree is the dividend degree minus divisor degree.
- The identity dividend = divisor × quotient + remainder can check the result.
SECTION 03
Factorising cubic polynomials
Test likely rational roots, obtain a linear factor, then divide to get a quadratic. Factorise the quadratic if possible; otherwise use the quadratic formula.
| Given information | Immediate test |
|---|---|
| x − a is a factor | substitute x = a |
| x + a is a factor | substitute x = −a |
| integer constant term | test integer divisors of the constant |
| remainder r on division by x − a | set f(a) = r |
SECTION 04
Cubic equations and graph links
A fully factorised cubic equals zero when any factor equals zero. Repeated factors create repeated roots, while the sign of the leading coefficient controls end behaviour.
DETAILED EXPLANATION
- List all real solutions after setting each factor to zero.
- A cubic with three distinct linear factors crosses the axis at each simple root.
- A repeated even-multiplicity factor touches rather than crosses.
- Substitute roots back if coefficients or divisions were complicated.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- The remainder from division by x − a is f(a).
- A zero remainder establishes a factor.
- Polynomial division reduces the degree systematically.
- One cubic root leads to a quadratic factor.
- A cubic equation is solved only after every factor has been considered.
QUICK REVISION CHECKLIST
Can you do each of these without your notes?
- apply the remainder theorem
- use the factor theorem in both directions
- divide a polynomial by a linear expression
- factorise and solve cubic equations