Verify common-factor and square-difference products lesson | Year 10 Mathematics (Quadratic and non-linear graphs, Common factors and difference of squares)

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

Decide whether this proposed factorisation is equivalent and complete

12x+18=2(6x+9)

\(\displaystyle 2(6x+9)=12x+18\)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
\(\displaystyle 6(2x+3)\)The proposal is equivalent but incomplete. This is the complete integer-coefficient factorisation

A true equality can still be unfinished

✕\(\displaystyle 6x+12=2(3x+6)\)✓\(\displaystyle 6x+12=6(x+2)\)

The first equality is true, but the bracket retains a common factor 3.

Losing a constant sign

✕\(\displaystyle x^2-25=(x+5)^2\)✓\(\displaystyle x^2-25=(x+5)(x-5)\)

The factors need opposite constant signs so their cross-products cancel

Dividing only one term

✕\(\displaystyle 6x+9=3(2x+9)\)✓\(\displaystyle 6x+9=3(2x+3)\)

Every term must be divided by the outside factor

introVerify a common-factor product and read its ordered coefficients

standardVerify a difference-of-squares product and evaluate it exactly

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 10 Mathematics, chapter 10: Quadratic and non-linear graphs