# What is the standard form of  y= (-7x-9)^3-(2x-1)^2?

Nov 24, 2017

$y = - 343 {x}^{3} - 1327 {x}^{2} - 1697 x - 730$

#### Explanation:

This is a really ugly problem, but the binomial theorem helps a lot. The binomial theorem lets you raise binomials to high powers without exhausting a lot of time.

We'll tackle the cube first.

(x+y)^n=sum_(r=0)^n""^nC_rx^(n-r)y^r

Here, we can substitute our values.

(-7x+ -9)^3=sum_(r=0)^3""^3C_r(-7x)^(3-r)(-9)^r

Using the binomial theorem, we can find that:

${\left(- 7 x - 9\right)}^{3} = - 343 {x}^{3} + 3 \cdot 49 {x}^{2} \cdot - 9 + 3 \cdot - 7 x \cdot 81 + - 729$

${\left(- 7 x - 9\right)}^{3} = - 343 {x}^{3} - 1323 {x}^{2} - 1701 x - 729$

The second part is easier:

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

Now we add the two together:

$y = - 343 {x}^{3} - 1327 {x}^{2} - 1697 x - 730$