Factorising a Trinomial Revision Notes for Grade 12 NSC Matric Mathematics

Factorising a Trinomial

What is a trinomial?

A trinomial is a polynomial expression that contains exactly three terms. These expressions typically follow the standard form:

$ax^2 + bx + c$

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero.

Factorising a trinomial means breaking it down and expressing it as the product of two binomial expressions. This process is the reverse of expanding brackets and is a fundamental skill in algebra that helps solve quadratic equations and simplify complex expressions.

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Factorisation is essentially the inverse operation of expanding brackets. If you can expand $(2x + 3)(x + 1)$ to get $2x^2 + 5x + 3$ , then factorising $2x^2 + 5x + 3$ should give you back $(2x + 3)(x + 1)$ .

Steps for factorising a trinomial

The factorisation process follows a systematic five-step method that works reliably for most trinomials.

Step 1: Multiply the coefficients

Begin by identifying the coefficient of the x² term (first term) and the constant term (last term). Multiply these two numbers together to get a product that you'll use in the next step.

For example, when factorising $3x^2 + 11x + 6$ :

Coefficient of $x^2$ = 3
Constant term = 6
Product = $3 \times 6 = 18$

Step 2: Find two numbers

You need to find two numbers that satisfy two conditions simultaneously:

They multiply together to give the product from Step 1
They add together to give the middle coefficient (the coefficient of the x term)

Using our example with the product 18 and middle coefficient 11:

Find two numbers that multiply to 18 and add to 11
The answer is 9 and 2 (since $9 \times 2 = 18$ and $9 + 2 = 11$ )

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This is often the most challenging step. If you cannot find two numbers that satisfy both conditions, the trinomial may not be factorisable using integers.

Step 3: Rewrite the middle term

Replace the middle term of the trinomial by splitting it into two terms using the numbers you found in Step 2.

Continuing with our example:

Original: $3x^2 + 11x + 6$
Rewritten: $3x^2 + 9x + 2x + 6$

Step 4: Group and factorise

Group the four terms into two pairs and factorise each pair by taking out common factors.

From our example:

Group: $(3x^2 + 9x) + (2x + 6)$
Factorise each group: $3x(x + 3) + 2(x + 3)$

Notice that both groups now contain the same bracket $(x + 3)$ .

Step 5: Factorise the common bracket

Take out the common bracket to complete the factorisation.

Final step for our example:

Result: (3x + 2)(x + 3)

Worked examples

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Worked Example 1: Factorise $4x^2 + 9x - 13$

Step 1: Multiply the coefficients $4 \times (-13) = -52$

Step 2: Find two numbers that multiply to -52 and add to 9 The numbers are 13 and -4 (since $13 \times (-4) = -52$ and $13 + (-4) = 9$ )

Step 3: Rewrite the middle term

$4x^2 - 4x + 13x - 13$

Step 4: Group and factorise

$(4x^2 - 4x) + (13x - 13)$

$= 4x(x - 1) + 13(x - 1)$

Step 5: Factor out the common bracket

$= (4x + 13)(x - 1)$

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Worked Example 2: Factorise $8x^2 - 18x + 9$

Step 1: Multiply the coefficients $8 \times 9 = 72$

Step 2: Find two numbers that multiply to 72 and add to -18 The numbers are -12 and -6 (since $(-12) \times (-6) = 72$ and $(-12) + (-6) = -18$ )

Step 3: Rewrite the middle term

$8x^2 - 12x - 6x + 9$

Step 4: Group and factorise

$(8x^2 - 12x) - (6x - 9)$

$= 4x(2x - 3) - 3(2x - 3)$

Step 5: Factor out the common bracket

$= (4x - 3)(2x - 3)$

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Worked Example 3: Factorise $x^2 + 7x + 12$

Step 1: Multiply the coefficients $1 \times 12 = 12$

Step 2: Find two numbers that multiply to 12 and add to 7 The numbers are 3 and 4 (since $3 \times 4 = 12$ and $3 + 4 = 7$ )

Step 3: Rewrite the middle term

$x^2 + 3x + 4x + 12$

Step 4: Group and factorise

$(x^2 + 3x) + (4x + 12)$

$= x(x + 3) + 4(x + 3)$

Step 5: Factor out the common bracket

$= (x + 3)(x + 4)$

bookmarkSummary

Key Steps to Remember:

When factorising trinomials, always follow this systematic approach:

Multiply the coefficient of the first term by the constant term
Find two numbers that multiply to give this product and add to give the middle coefficient
Rewrite the middle term by splitting it using these two numbers
Group the terms into two pairs and factorise each pair
Factor out the common bracket to complete the factorisation

Common exam tips

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Essential Exam Tips:

Always check your answer by expanding the brackets to verify you get back to the original trinomial
Be careful with negative signs, especially when the constant term is negative
If the coefficient of $x^2$ is 1, the process becomes simpler as you only need to find factors of the constant term
Practice identifying when a trinomial cannot be factorised (when no suitable pair of numbers exists)

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Remember These Key Points:

A trinomial has exactly three terms and follows the form $ax^2 + bx + c$
Factorising means expressing the trinomial as a product of two binomials
The systematic five-step method works for most trinomials you'll encounter
Always multiply the first and last coefficients to find your target product
Check your final answer by expanding to ensure it matches the original expression

Factorising a Trinomial (Grade 12 NSC Matric Mathematics): Revision Notes