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2 Algebra
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Stage 5.1-2

2.07 Factorising

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Factorise algebraic expressions

The distributive law says that for any numbers A, B, and C, A\left(B+C\right)=AB+AC. The reverse of expanding algebraic expressions is called factorising. Factorising an algebraic expression means writing the expression with any common factors between the terms taken outside of the brackets.

It is helpful to find the highest common factor of the coefficients before we factorise the expression. Otherwise we might end up having to factorise a second time.

Examples

Example 1

Factorise the following expression by taking out the highest common factor:42x-x^2

Worked Solution
Create a strategy

Factorise using the law, AB+AC=A\left(B+C\right).

Apply the idea

The coefficients of the terms do not have any common factors other than 1, so we cannot factorise out any numbers. However, both terms have at least one x so we can factorise that out.

\displaystyle 42x-x^2\displaystyle =\displaystyle x\times 42 +x \times Write the terms as products of x
\displaystyle =\displaystyle x\left(42-x\right)Factorise

Example 2

Factorise the expression -2s-10.

Worked Solution
Create a strategy

Factorise using the law, AB+AC=A\left(B+C\right).

Apply the idea
\displaystyle -2s-10\displaystyle =\displaystyle -2\times s+\left(-2\right)\times 5Write the terms as products of -2
\displaystyle =\displaystyle -2\left(s+5\right)Factorise
Reflect and check

It is important to be careful about factorising a negative because the sign of each term will change. If the leading term in an expression is negative, then we should factorise the negative out.

Example 3

Factorise the following expression: 2t^{2}k^{7}+18 t^{9}k^{9}

Worked Solution
Create a strategy

Factorise using the law, AB+AC=A\left(B+C\right).

Apply the idea

The coefficients have a common factor of 2, and both terms have factors of t^2 and k^7. So we can factorise 2t^2 k^7 out.

\displaystyle 2t^{2}k^{7}+18t^{9}k^{9}\displaystyle =\displaystyle 2t^{2}k^{7}\times 1+ 2t^{2}k^{7}\left(9t^{7}k^{2} \right)Write the terms as products of 2x^{2}y^{7}
\displaystyle =\displaystyle 2t^{2}k^{7}\left(1+9t^{7}k^{2}\right)Factorise
Idea summary

We can use the distributive law to factorise an algebraic expression like AB+AC. This means means writing the expression with any common factors between the terms taken outside of the brackets: AB+AC=A(B+C) Factorising is the reverse of expanding.

Outcomes

MA5.2-6NA

simplifies algebraic fractions, and expands and factorises quadratic expressions

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