# 4.2 Factorisation  (Page 2/2)

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There are some tips that you can keep in mind:

• If $c$ is positive, then the factors of $c$ must be either both positive or both negative. The factors are both negative if $b$ is negative, and are both positive if $b$ is positive. 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, multiply out your brackets again just to make sure it really works.

Find the factors of $3{x}^{2}+2x-1$ .

1. The quadratic is in the required form.

2. $\left(\phantom{\rule{1.em}{0ex}}x\phantom{\rule{2.em}{0ex}}\right)\left(\phantom{\rule{1.em}{0ex}}x\phantom{\rule{2.em}{0ex}}\right)$

Write down a set of factors for $a$ and $c$ . The possible factors for $a$ are: (1,3). The possible factors for $c$ are: (-1,1) or (1,-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.

 Option 1 Option 2 $\left(x-1\right)\left(3x+1\right)$ $\left(x+1\right)\left(3x-1\right)$ $3{x}^{2}-2x-1$ $3{x}^{2}+2x-1$
3. $\begin{array}{ccc}\hfill \left(x+1\right)\left(3x-1\right)& =& x\left(3x-1\right)+1\left(3x-1\right)\hfill \\ & =& \left(x\right)\left(3x\right)+\left(x\right)\left(-1\right)+\left(1\right)\left(3x\right)+\left(1\right)\left(-1\right)\hfill \\ & =& 3{x}^{2}-x+3x-1\hfill \\ & =& {x}^{2}+2x-1.\hfill \end{array}$
4. The factors of $3{x}^{2}+2x-1$ are $\left(x+1\right)$ and $\left(3x-1\right)$ .

## Factorising a trinomial

1. Factorise the following:
 (a) ${x}^{2}+8x+15$ (b) ${x}^{2}+10x+24$ (c) ${x}^{2}+9x+8$ (d) ${x}^{2}+9x+14$ (e) ${x}^{2}+15x+36$ (f) ${x}^{2}+12x+36$
2. Factorise the following:
1. ${x}^{2}-2x-15$
2. ${x}^{2}+2x-3$
3. ${x}^{2}+2x-8$
4. ${x}^{2}+x-20$
5. ${x}^{2}-x-20$

3. Find the factors for the following trinomial expressions:
1. $2{x}^{2}+11x+5$
2. $3{x}^{2}+19x+6$
3. $6{x}^{2}+7x+2$
4. $12{x}^{2}+8x+1$
5. $8{x}^{2}+6x+1$

4. Find the factors for the following trinomials:
1. $3{x}^{2}+17x-6$
2. $7{x}^{2}-6x-1$
3. $8{x}^{2}-6x+1$
4. $2{x}^{2}-5x-3$

## Factorisation by grouping

One other method of factorisation involves the use of common factors. We know that the factors of $3x+3$ are 3 and $\left(x+1\right)$ . Similarly, the factors of $2{x}^{2}+2x$ are $2x$ and $\left(x+1\right)$ . Therefore, if we have an expression:

$2{x}^{2}+2x+3x+3$

then we can factorise as:

$2x\left(x+1\right)+3\left(x+1\right).$

You can see that there is another common factor: $x+1$ . Therefore, we can now write:

$\left(x+1\right)\left(2x+3\right).$

We get this by taking out the $x+1$ and seeing what is left over. We have a $+2x$ from the first term and a $+3$ from the second term. This is called factorisation by grouping .

Find the factors of $7x+14y+bx+2by$ by grouping

1. There are no factors that are common to all terms.

2. 7 is a common factor of the first two terms and $b$ is a common factor of the second two terms.

3. $7x+14y+bx+2by=7\left(x+2y\right)+b\left(x+2y\right)$
4. $x+2y$ is a common factor.

5. $7\left(x+2y\right)+b\left(x+2y\right)=\left(x+2y\right)\left(7+b\right)$
6. The factors of $7x+14y+bx+2by$ are $\left(7+b\right)$ and $\left(x+2y\right)$ .

## Factorisation by grouping

1. Factorise by grouping: $6x+a+2ax+3$
2. Factorise by grouping: ${x}^{2}-6x+5x-30$
3. Factorise by grouping: $5x+10y-ax-2ay$
4. Factorise by grouping: ${a}^{2}-2a-ax+2x$
5. Factorise by grouping: $5xy-3y+10x-6$

what is the stm
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scanning tunneling microscope
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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absolutely yes
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for teaching engĺish at school how nano technology help us
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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