Factorisation of Cubic Expressions

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Factorising quadratic expressions is comparatively easy. If the coefficient ofis one you can often find the factors by inspection. For example, to factorisefind 2 numbers that add or take away to give 3 and multiply to give 18. By inspection we obtain 6 and 3. Then we can factorise:

When we try and factorise a cubic we can start by finding common factors. This may reduce the problem to one of factorising a quadratic:

The expression inside the brackets now factorises by inspection: find two numbers that add or take away to give -5 and multiply to give 6. We obtain -2 and -3. Hence,

If we can’t reduce the problem to factorising a quadratic by inspection, then things get a little more involved. Consider how to factorise a quadratic where the coefficient ofis not one. For example,

Multiply the coefficient of2 by the constant term, 5 to get 10. Now look for the two factors of 10 that add to give the coefficient of7. The two factors are 2 and 5. Now

Example: Factorise the cubic expression

Factorise first with the common factor 3x to give

To factorise the quadratic in the brackets, multiply the coefficient of2 by the constant term, 7 to get 14, then find the factors of 14 that add to give 9. The answer is 2 and 7. Hence

hence the cubic factorises as

Example: Factorise the cubic expression

Factorise first with the common factor 2x to give

To factorise the quadratic in the brackets, multiply the coefficient of2 by the constant term, 9 to get 18, then find the factors of 18 that add to give 9. The answer is 3 and 6. Hence

hence the cubic factorises as

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