3x + 15 can be factorised because 3x and 15 have a common factor of 3.
3 times x is 3x, and 3 times 5 is 15.
The original expression is therefore factorised as follows,
$$3(x\;+\;5)$$
Another example is 2x + 14.
2 times x is 2x, and 2 times 7 is 14.
The original expression is therefore factorised as follows,
$$2(x\;+\;7)$$
An Example of Factorising a Quadratic Expression
To factorise the following quadratic expression $$3x^2\; + \;10x\; -\; 8\;$$begin by finding two numbers that multiply to get the product of the coefficient of the first term, 3, and the last term, -8, and that also add up to the coefficient of the middle term, 10. 3 multiplied by -8 is -24. And 12 multiplied by -2 is also -24, and they also add together to make 10. Using 12 and -2 as the coefficients of the middle term, in the place of 10, we can then make the following expression,
$$3x^2\;+12x\; -2x\;-\;8$$
We try to find a common factor between the two halves which can be expressed in brackets, (the two middle terms may need to be swapped), as follows,
3x(x +4), for the first half of the expression, -2(x + 4), for the second half of the expression.
The terms inside the brackets should be the same in each instance. In this case (x + 4). So now we have,
3x(x +4) -2(x + 4)
Then to get the fully factorised answer, we put the terms which are inside the brackets inside the first set of brackets, and the terms which are outside the brackets inside the second set of brackets, as follows,
$$(x\;+\;4)(3x\;-\;2)$$
Finally, we check that these are the correct factors by expanding the answer, to see if the result is the original expression, by multiplying and adding the terms of the factors together.
The first and third term together: x times 3x = 3x squared
The first and fourth term together: x times -2 = -2x
The second and third term together: 4 times 3x = 12x
The second and fourth term together: 4 times -2 = -8
Adding all of these terms together, we get the original expression,$$3x^2 \;-2x \;+\; 12x\; -\; 8$$$$3x^2 \;+ \;10x \;- \;8$$
Therefore, the factorised answer,$$(x\;+\;4)(3x\;-\;2)$$is correct.
Expanding
Expanding is the reverse of factorising.
You can expand 4(x + 2) by multiplying both of the terms inside the brackets by the term on the outside the brackets.
4 times x is 4x, and 4 times 2 is 8.
The original expression of 4(x + 2) is therefore expanded as follows,
$$4x\;+\;8$$
Another Example
You can expand (x-4)(x + 2) by multiplying each of the terms inside one set of brackets by each of the terms in the other set of brackets.
First, multiply the first term, x, by the third term, x, to get, x squared,
Next, multiply the first term, x, by the fourth term, 2, to get, 2x,
Next, multiply the second term, -4, by the third term, x, to get, -4x.
Lastly multiply the second term, -4, by the fourth term, 2, to get, -8,
Adding all of these results together we get,$$x^2 \;+ \;2x\; -\; 4x \;- \;8$$
The expanded version of the original expression of (x-4)(x + 2) is therefore,$$x^2\;-\;2x\;-8\;$$
