How do you factorise fully 3 a^3 b + 12 a^2 b^2 + 9 a^5 b^3?

WORKING OUT

3a³b+12a²b²+9a⁵b³

1) 3(a³b+4a²b²+3a⁵b³)

2) 3a²(ab+4b²+3a³b³)

3) Fully factorised: 3a²b(a+4b+3a³b²)

BRIEFLY EXPLAINED

Factorising is the process of taking out any factors that are common to all parts of an expression and putting them outside of a bracket.

eg. 4x²+2x becomes 2x(2x+1), as 2x is common to all parts of the expression.

When factorising long expressions like 3a³b+12a²b²+9a⁵b³, the key is to work step by step.

1) What numbers does each part have in common? (eg what is the biggest number you can divide each part by and be left with integers (whole numbers, not fractions))

Both 3, 12 and 9 can be divided by 3 (you know this is the biggest common factor, as 3 cannot be divided by anything bigger and remain an integer), therefore you can "take 3 out of the brackets", like this:

3(a³b+4a²b²+3a⁵b³).

2) Now, the same process has to be done for the letters in this new expression in the brackets.

You can see that all parts of the equation can be divided by "a" twice (as a² is the same as a multiplied by a, and therefore a² divided by a is a, and a divided by a is 1) before creating a fraction. Both other parts of the expression can be divided by "a" twice without becoming fractions. Therefore a² can be taken out of the brackets, as it is the highest value of "a's" that each part of the expression can be divided by.

This would create:

 3a²(ab+4b²+3a³b³).

3) Step 3 is sorting the "b"s out. This can be done in the same way as the "a"s.

The greatest number of "b"s that all parts of the expression can be divided by is just b, therefore b can be taken out of the expression to leave:

3a²b(a+4b+3a³b²).

FACTORISATION IN GENERAL

Factorisation is the process of putting an expression into brackets by taking the factors that are common to all parts of the expression and taking them outside of the brackets.

For example, factorising 4x²+2x. You can divide both parts of the expression by 2 and still be left with no fractions (only integers or whole numbers), therefore you can "take two out of the brackets", like this:

2(2x²+x)

As you can see this new expression still has the same value as the original expression, however it is partially factorised.

In factorisation, both numbers and letters have to be taken into account, so the next step is to look at what letters each part of the new expression has in common. You can see that you can divide both parts by x and still be left with an integer, or whole number, (this is because x² is just the same as x multiplied by x). Therefore you can "take x out of the brackets" as well to leave:

2x(2x+1)

(remember: x divided by x is 1).

This is the simplest form that 4x²+2x can be simplified to, as there are no longer any common factors between the parts of the expression.

This process of factorisation can be applied to longer expressions, like 3a³b+12a²b²+9a⁵b³ as well.