Verify a complete factorisation
A factorisation must preserve every term. Factor order may change, but a numbered table uses the order explicitly requested
- Preserve the whole polynomial Expand the proposed factors and compare every signed coefficient with the original
- Equality and completeness are different An equivalent product can still have a factor left inside its brackets. Fully factorised products have no shared integer-coefficient factor left to remove
- Factor order can change Multiplication allows factors to swap places. In a numbered coefficient table, follow the specified factor order and signs
- Substitution is a check A disagreement disproves a proposal. Agreement at one value alone is insufficient to prove an identity
Worked example
Decide whether this proposed factorisation is equivalent and complete
- \(\displaystyle 2(6x+9)=12x+18\)Multiplication confirms equivalence
- \(\displaystyle 6x+9=3(2x+3)\)A common factor 3 remains inside the bracket
- \(\displaystyle 2\times3=6\)Combine the numerical outside factors
- \(\displaystyle 6(2x+3)\)This is a complete integer-coefficient factorisation
Common mistakes
Difficulty levels
Questions about this topic
Can I swap the factors?+
Yes, the product is unchanged. For the coefficient table, use the left-to-right factor order and signs of the correct product in part (a)
Why can an equivalent product be incomplete?+
Its bracket may still contain a common factor or an applicable difference of perfect squares
Do negative inputs invalidate a polynomial identity?+
No. An algebraic polynomial identity holds for every real input. Physical length models have separate positive-length conditions
Does the table include a numerical outside factor?+
Yes. If the displayed product begins with a numerical factor, that is factor 1. Missing powers have coefficient zero
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 8: Algebraic expressions and fractions