General Procedure For Factorising a Trinomial
General Procedure For Factorising a Trinomial
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Take out any common factor in the coefficients of the terms of the expression to obtain an expression of the form \(a{x}^{2} + bx + c\) where \(a\), \(b\) and \(c\) have no common factors and \(a\) is positive.
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Write down two brackets with an \(x\) in each bracket and space for the remaining terms:
\[(x \qquad )(x \qquad)\] -
Write down a set of factors for \(a\) and \(c\).
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Write down a set of options for the possible factors for the quadratic using the factors of \(a\) and \(c\).
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Expand all options to see which one gives you the correct middle term \(bx\).
Important:
If \(c\) is positive, then the factors of \(c\) must be either both positive or both negative. If \(c\) is negative, it means only one of the factors of \(c\) is negative, the other one being positive. Once you get an answer, always multiply out your brackets again just to make sure it really works.
The following video summarises how to factorise expressions and shows some examples.
Example
Question
Factorise: \(3x^2 + 2x - 1\).
Check that the quadratic is in required form \(a{x}^{2} + bx + c\)
Write down a set of factors for \(a\) and \(c\)
\[(x \qquad )(x \qquad )\]The possible factors for \(a\) are: 1 and 3
The possible factors for \(c\) are: \(-\text{1}\) and 1
Write down a set of options for the possible factors of the quadratic using the factors of \(a\) and \(c\). Therefore, there are two possible options.
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Option 1 |
Option 2 |
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\((x-1)(3x+1)\) |
\((x+1)(3x-1)\) |
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\(3{x}^{2}-2x-1\) |
\(3{x}^{2}+2x-1\) |
Check that the solution is correct by multiplying the factors
\begin{align*} (x + 1)(3x - 1) & = 3{x}^{2} - x + 3x - 1\\ & = 3{x}^{2} + 2x - 1 \end{align*}Write the final answer
\(3{x}^{2} + 2x - 1 = (x + 1)(3x - 1)\)
This lesson is part of:
Algebraic Expressions Overview