Take out the greatest numerical and variable factor from every term.
Core & Extended · Syllabus C2.2, E2.2
Notes and key methods
Factorising reverses expansion. The greatest common numerical factor uses a numerical HCF, while a common variable power uses its smallest exponent among the terms. Divide every term by the extracted factor to find what stays inside the bracket.
Expand the answer to check every coefficient and exponent. A factorisation is not complete when the remaining bracket still has a common factor. Taking out a negative factor changes every sign in its bracket.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Extracting a common algebraic factor
Consider \(E=10 x^{3} y + 15 x^{2} y^{2}\).
Factorise E fully.
Find the exponent of x in the greatest common monomial factor.
After taking out \(5x^{2}y\), find the coefficient of y inside the bracket.
Evaluate E at \(\displaystyle x=2, y=3\).
Show the reasoning
(a)
The numerical HCF is 5. The common variable factor is \(x^{2}y\). Dividing both terms by their product gives \(2 x + 3 y\).
Answer: \(5 x^{2} y \left(2 x + 3 y\right)\)
(b)
Use the HCF and the smallest variable exponents, dividing every original term by the complete common factor.
\(2\)
Answer: \(2\).
(c)
Use the HCF and the smallest variable exponents, dividing every original term by the complete common factor.
\(3\)
Answer: \(3\).
(d)
Use the HCF and the smallest variable exponents, dividing every original term by the complete common factor.
\(780\)
Answer: \(780\).
Example 2 · Extracting a common algebraic factor
Consider \(E=18 x^{4} y + 24 x^{3} y^{2}\).
Factorise E fully.
Find the exponent of x in the greatest common monomial factor.
After taking out \(6x^{3}y\), find the coefficient of y inside the bracket.
Evaluate E at \(\displaystyle x=2, y=3\).
Show the reasoning
(a)
The numerical HCF is 6. The common variable factor is \(x^{3}y\). Dividing both terms by their product gives \(3 x + 4 y\).
Answer: \(6 x^{3} y \left(3 x + 4 y\right)\)
(b)
Use the HCF and the smallest variable exponents, dividing every original term by the complete common factor.
\(3\)
Answer: \(3\).
(c)
Use the HCF and the smallest variable exponents, dividing every original term by the complete common factor.
\(4\)
Answer: \(4\).
(d)
Use the HCF and the smallest variable exponents, dividing every original term by the complete common factor.
\(2592\)
Answer: \(2592\).
Example 3 · Extracting a common algebraic factor
Consider \(E=2 x^{2} y + 4 x y^{2}\).
Factorise E fully.
Find the exponent of x in the greatest common monomial factor.
After taking out \(2x^{1}y\), find the coefficient of y inside the bracket.
Evaluate E at \(\displaystyle x=2, y=3\).
Show the reasoning
(a)
The numerical HCF is 2. The common variable factor is \(x^{1}y\). Dividing both terms by their product gives \(x + 2 y\).
Answer: \(2 x y \left(x + 2 y\right)\)
(b)
Use the HCF and the smallest variable exponents, dividing every original term by the complete common factor.
\(1\)
Answer: \(1\).
(c)
Use the HCF and the smallest variable exponents, dividing every original term by the complete common factor.
\(2\)
Answer: \(2\).
(d)
Use the HCF and the smallest variable exponents, dividing every original term by the complete common factor.
\(96\)
Answer: \(96\).
Common mistakes
Keep intermediate values exact until the final step.