# What is the standard form of y= (2x+3x^2)(x+3) -(x-2)^3?

Dec 14, 2015

$y = 2 {x}^{3} + 17 {x}^{2} - 6 x + 8$

#### Explanation:

To answer this question, you will have to simplify the function. Begin by using the FOIL Method to multiply the first term:

$\left(2 x + 3 {x}^{2}\right) \left(x + 3\right) = 2 x \cdot x + 2 x \cdot 3 + 3 {x}^{2} \cdot x + 3 {x}^{2} \cdot 3$

Simplifying this yields:

$3 {x}^{3} + 11 {x}^{2} + 6 x$

We now have the first term simplified. To simplify the second term, we can use the
Binomial Theorem, a useful tool when working with polynomials. One of the main points of the theorem is that the coefficients of an expanded binomial can be determined using a function called the choose function. The specifics of the choose function are more of a probability concept, so there's no need to go into it right now.

However, a simpler way to use the Binomial Theorem is
Pascal's Triangle. The numbers in Pascal's Triangle for a certain row number will correspond to the coefficients of the expanded binomial for that row number. In the case of cubing, the third row is $1 , 3 , 3 , 1$, so the expanded binomial would be:

${\left(a + b\right)}^{3} = 1 {a}^{3} + 3 {a}^{2} b + 3 a {b}^{2} + 1 {b}^{3}$

Notice how we decrease the power of $a$ and increase the power of $b$ as we move down the row. Evaluating this formula with the second term, ${\left(x - 2\right)}^{3}$, yields:

${\left(x - 2\right)}^{3} = {x}^{3} + 3 {x}^{2} \left(- 2\right) + 3 x {\left(- 2\right)}^{2} + {\left(- 2\right)}^{3}$

Simplifying gives us:

${x}^{3} - 6 {x}^{2} + 12 x - 8$

To simplify, we can subtract the second term from the first:

$3 {x}^{3} + 11 {x}^{2} + 6 x - \left({x}^{3} - 6 {x}^{2} + 12 x - 8\right) = 2 {x}^{3} + 17 {x}^{2} - 6 x + 8$

Standard form means that the terms of the polynomial are ordered from highest degree to lowest. Because this has already been done, your final answer is:

$y = 2 {x}^{3} + 17 {x}^{2} - 6 x + 8$