AeroLab Reading room

MATH / A2 level

Advanced algebra and numerical methods

Polynomial division and the factor theorem reduce higher-degree equations. Partial fractions rewrite rational expressions into forms useful for integration.

Learning objectives

The big idea

Polynomial division and the factor theorem reduce higher-degree equations. Partial fractions rewrite rational expressions into forms useful for integration.

Iterative methods approximate roots. Convergence depends on the function and starting value; report accuracy based on interval or convergence evidence rather than screen digits alone.

Equations and relationships

f(a)=0 implies (x−a) is a factor

Factor theorem for polynomials.

Worked example

For f(x)=x³−8, is x−2 a factor?

Follow the solution

f(2)=8−8=0, so the factor theorem applies.

Investigate and explain

Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.

Use the prediction activity below; a local model appears only where it supports this topic.

Open the interactive investigation

Check your understanding

1. What must be done before decomposing an improper rational fraction?

  1. Take a square root
  2. Polynomial division
  3. Set the denominator to zero only
Check the answer and reasoning

Polynomial division

Division separates a polynomial part from a proper rational fraction.

2. For f(x)=x³−8, is x−2 a factor?

  1. No
  2. Only if x=0
  3. Yes
Check the answer and reasoning

Yes

f(2)=8−8=0, so the factor theorem applies.

3. Does an iteration that changes little prove the exact root was found?

  1. No
  2. Yes
  3. Only for a positive root
Check the answer and reasoning

No

Small numerical changes need convergence and residual checks; finite precision can also stall iteration.