Factorise common factors and differences of squares practice questions | Year 10 Mathematics (Quadratic and non-linear graphs, Common factors and difference of squares)

Common factors and square differences

Factorisation reverses expansion. Check equivalence and then completion; use integer-coefficient factors in this section

Divide every term A common factor multiplies the whole bracket. Include shared variables, using their lowest power \(\displaystyle 18x^2+30x=6x(3x+5)\)
Carry negative signs Taking out a negative factor reverses every sign inside \(\displaystyle -12x^2+18x=-6x(2x-3)\)
Difference of squares Identify both square roots. Opposite signs cancel the cross-products \(\displaystyle a^2-b^2=(a-b)(a+b)\)
Finish the remaining bracket An equivalent intermediate product may still factor further. Expand to verify the final product \(\displaystyle 2(x^2-9)=2(x-3)(x+3)\)

Factorise completely, keeping the outside factor

18x^2-50

\(\displaystyle \begin{aligned}&\operatorname{HCF}(18,50)\\&=2\end{aligned}\)There is no shared variable because the second term is constant
\(\displaystyle 2(9x^2-25)\)Divide both terms by 2.
\(\displaystyle 9x^2=(3x)^2\)Identify the first square
\(\displaystyle 25=5^2\)Identify the second square
\(\displaystyle 2(3x-5)(3x+5)\)Retain 2 and use opposite signs
\(\displaystyle 2(9x^2-25)=18x^2-50\)Expand to check every term

Equivalent but unfinished

✕\(\displaystyle 18x^2+30x=3x(6x+10)\)✓\(\displaystyle 18x^2+30x=6x(3x+5)\)

The first equality is true, but a common factor 2 remains in the bracket

A square is not a square difference

✕\(\displaystyle x^2-49=(x-7)^2\)✓\(\displaystyle x^2-49=(x-7)(x+7)\)

Squaring x − 7 introduces −14x and gives +49.

Keep the outside factor

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

The outside factor multiplies the entire expression

introExtract numerical and variable common factors

standardFactor a difference of squares

challengeRemove a common factor before a difference of squares

examDistinguish complete factors from equivalent intermediate products

Must I take the highest common factor first?+

It is efficient, but not compulsory. Smaller common factors are valid steps if you continue until the factorisation is complete

Can I take out a negative factor?+

Yes. Divide every term by that negative factor and retain all resulting signs

Can the two linear factors swap places?+

Yes. Their product is unchanged. For a numbered input template, use the order and sign convention shown

Why does the picture use positive lengths?+

Physical areas need positive lengths. The identity follows by expansion and holds for all real algebraic inputs

What about monic and non-monic trinomials?+

The next section develops those factorisations. This section focuses on common factors and differences of squares

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