Lesson 6, Topic 3
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# Factorisation of Trinomials of the Form x^2 + bx + c

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A trinomial is an algebraic expression containing three terms.

For example, ax2 + b  + c is a trinomial because it has three terms ie. ax2, bx, c  when a = 1  this expression becomes  x2 + b x  + c,  a simple trinomial.

Some quadratic expression can be factorised by splitting the middle term.

Therefore, to factorize a trinomial of the form ax2 + bx + c, look for pairs of factors of the constant terms c that adds up to b i.e. the coefficient of x.

Example 1

a. Factorise   x2 + 7x  + 10

i. First multiply The first and the last term i.e. a × c

ii. Look for the factors of 10x2, two factors when they are multiplied give 10x2 and when added they give 7x, which is the middle term.

Factors of 10x2 that add up to 7x are 2x and 5x

$$\scriptsize 2x \: \times \: 5x = 10x^2$$

$$\scriptsize 2x \: + \: 5x = 7x$$

So replace 7x with 2x + 5x

=  x2 + 2x  + 5x + 10

Factorise by grouping

(x2 + 2x ) + (5x + 10)

x (x + 2 ) + 5 (x + 2)

= (x + 5 )(x + 2)

b.   x2 + 3x  + 2

Factors of 2x2 that add up to 3x are 2x and 1x

So replace 3x with 2x + 1x

=  x2 + 2x  + 1x + 2

Factorise by grouping

(x2 + 2x ) + (x  + 2)

x(x + 2) + 1(x  + 2)

(x  + 1)(x  + 2)

c.   x2 + 8x  + 15

=  x2 + 3x + 5x + 15

(x 2 + 3x) + (5x + 15)

x(x + 3) + 5(x + 3)

(x + 5)(x + 3)

d. y2 + 9y + 18

= y2 + 8y + 1y + 18

(y2 + 8y) + (1y + 8)

y (y + 8) +1(y + 8)

= (y + 1)( y + 8)

e.  2x2 + 13x + 6

=  2x2 + 12x + x + 6

(2x2 + 12x ) + (x  + 6)

2x (x + 6) + 1(x + 6)

(2x + 1)(x + 6)

Further Examples

i. 4x2 – 3x – 22

= 4x2 + 8x – 11x – 22

(4x2 + 8x ) – (11x + 2)

4x (x + 2) – 11 (x  + 2)

= (4x – 11) (x  + 2)

ii. 2e2 – 3e + 1

2e2e – 2e + 1

(2e2e) – (2e – 1)

e(2e1) – 1(2e – 1)

= (e – 1)(2e – 1)

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