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Mathematics in Business and Economics

Other Important Principles in Algebra

Factorising

When factorising, we are attempting to break a number or a term into smaller components. The factors of 12, for example, are 1, 2, 3, 4, 6 and 12. These are all numbers that we can multiply by others to get 12. If we have an expression such as y² - 10y + 16 we can factorise it by finding two numbers which when added together give the middle number (10) but which when multiplied together give the last number (16). The factors of -10 and 16 are 2, 4 and 8. -2 x -8 gives us 16 and (-2) + (-8) = -10. This equation can be written in its factorised form of (y-2)(y-8). To check we can multiply out the brackets as follows:

Take the first term and multiply it by each term in the second bracket, then take the second term in the first bracket and multiply by the two terms in the second. Add them all together!

y x y = y², y x -8 = -8y
-2 x y = -2y and -2 x -8 = 16

So, when adding like terms together we get back to the original equation, y² -10y + 16.

Dividing Terms in Algebra

As in normal division, look to cancel down where appropriate. If the divisor is a letter representing a number, then the same principle still applies. An example may serve to illustrate the point more clearly.

Take the following equation:

(35y³ + 10y² - 15y) / 5y

We can divide 5y into 35y³, the answer being 7y². 5y goes into 10y² 2y times. 5y goes into -15y -3 times. This gives us the simplified equation 7y² + 2y - 3.

To take another example:

(21y²+ 14y - 70) / 7 = 3y² +2y - 10

In cases where the divisor will not go into the numerator, we can still simplify the equation but end up with a fraction as shown below:

(3y² + y - 9) / 3 = y² + (1/3)y - 3

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