Question

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

Answer 1

`y=-343x^3-1327x^2-1697x-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:

`(-7x-9)^3=-343x^3+3*49x^2*-9+3*-7x*81+ -729`

`(-7x-9)^3=-343x^3-1323x^2-1701x-729`

The second part is easier:

`-(2x-1)^2=-(4x^2-4x+1)=-4x^2+4x-1`

Now we add the two together:

`y=-343x^3-1327x^2-1697x-730`