# What is the standard form of  y=(2x + 1) (3x – 4) (2x – 1) ?

Jan 27, 2016

$y = 12 {x}^{3} - 16 {x}^{2} - 3 x + 4$

#### Explanation:

Visual inspection of the equation shows that it is a cubic function (there are 3 x's all with exponent 1). Hence we know that the standard form of the equation should appear this way:

$y = a {x}^{3} + b {x}^{2} + c x + d$

Generally in solving these types of questions, a possible approach would be expanding the equation. Sometimes this may seem tedious especially for longer equations however with a little patience you will be able to reach the answer. Of course it would also help if you know which terms to expand first to make the process less complicated.

In this case, you can choose which two terms you wish to expand first. So you can do either of the following

*Option 1
$y = \left(2 x + 1\right) \left(3 x - 4\right) \left(2 x - 1\right)$
$y = \left(6 {x}^{2} - 8 x + 3 x - 4\right) \left(2 x - 1\right)$
$y = \left(6 {x}^{2} - 5 x - 4\right) \left(2 x - 1\right)$

OR

*Option 2
$y = \left(2 x + 1\right) \left(2 x - 1\right) \left(3 x - 4\right)$ -> rearranging the terms
$y = \left(4 {x}^{2} - 1\right) \left(3 x - 4\right)$

Note that in Option 2 the product of $\left(2 x + 1\right) \left(2 x - 1\right)$ follows the general pattern of $\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$. In this case, the product is shorter and simpler than that of the 1st option. Therefore, although both options will lead you to the same final answer, it would be simpler and easier for you to follow the 2nd one.

Continuing with the solution from Option 2

$y = \left(4 {x}^{2} - 1\right) \left(3 x - 4\right)$
$y = 12 {x}^{3} - 16 {x}^{2} - 3 x + 4$

But if you still opt to do the 1st solution indicated above...

$y = \left(6 {x}^{2} - 5 x - 4\right) \left(2 x - 1\right)$
$y = 12 {x}^{3} - 6 {x}^{2} - 10 {x}^{2} + 5 x - 8 x + 4$
$y = 12 {x}^{3} - 16 {x}^{2} - 3 x + 4$

...it would still produce the same final answer