Common-factor factorisation
Factorisation reverses the distributive law. A complete common-factor factorisation removes the numerical HCF and every shared pronumeral power
- Use every term A proposed factor must divide every term, including constants
- Use the smallest common power For each pronumeral, take the minimum exponent present in every term. An absent pronumeral has exponent zero
- Distinguish complete from equivalent A smaller common factor may give a true identity but leave further factorisation inside the bracket
- Negative factors are allowed Taking out a negative factor changes the sign of every quotient. Either sign convention can represent the same polynomial
- Check by expansion Multiply every bracket term by the outside factor. The polynomial identity also holds where the outside factor is zero
Worked example
Factorise by taking out a common negative monomial
- \(\displaystyle \operatorname{HCF}(18,30)=6\)The positive numerical HCF is 6.
- \(\displaystyle -6y^{2}\)Take out negative 6y squared; the lowest y exponent is 2.
- \(\displaystyle \frac{-18y^{3}}{-6y^{2}}=3y\)First quotient, calculated for nonzero y.
- \(\displaystyle \frac{30y^{2}}{-6y^{2}}=-5\)Second quotient, calculated for nonzero y.
- \(\displaystyle -6y^{2}(3y-5)\)The complete factorisation. Expansion gives the original polynomial for every y, including zero
Common mistakes
Difficulty levels
Questions about this topic
Is an incomplete factorisation incorrect?+
It may still be equivalent to the original expression. If the question asks for complete factorisation, continue until no factor greater than 1 or shared pronumeral remains inside
Must the outside factor be positive?+
No. Opposite outside signs with opposite signs throughout the bracket give the same polynomial
Does factorisation exclude x = 0?+
A polynomial factorisation is an identity, so it remains valid at zero. Any intermediate division by x requires x to be nonzero; expansion verifies the final identity without that restriction
Can any two brackets be combined as a common factor?+
They must be identical, or first rewritten into an identical form with any sign change handled. Similar-looking brackets are not automatically the same factor
Does testing one value prove a factorisation?+
No. An incorrect expression can agree at a single value. Expand the proposed factors and compare every term
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 3: Algebra and indices