To factorise the expression [tex]\(12a^3 - 21a\)[/tex] completely, follow these steps:
1. Identify the common factor: Look at the coefficients and the variables in each term to identify a common factor.
- For the coefficients 12 and 21, the greatest common divisor is 3.
- For the variable terms [tex]\(a^3\)[/tex] and [tex]\(a\)[/tex], the lowest power of [tex]\(a\)[/tex] is [tex]\(a\)[/tex].
Thus, the common factor is [tex]\(3a\)[/tex].
2. Factor out the common factor: Divide each term in the expression by the common factor [tex]\(3a\)[/tex].
[tex]\[
3a \left( \frac{12a^3}{3a} \right) - 3a \left( \frac{21a}{3a} \right)
\][/tex]
Simplifying inside the parentheses:
[tex]\[
3a (4a^2) - 3a (7)
\][/tex]
This gives:
[tex]\[
3a (4a^2 - 7)
\][/tex]
3. Check for further factorisation: The term inside the parentheses, [tex]\(4a^2 - 7\)[/tex], should be checked if it can be factored further. In this case, [tex]\(4a^2 - 7\)[/tex] does not factorise further using real numbers (since it is not a difference of squares or any recognizable pattern that factors further).
Thus, the completely factorised form of the given expression [tex]\(12a^3 - 21a\)[/tex] is:
[tex]\[
3a(4a^2 - 7)
\][/tex]
