Question

What is the standard form of `y(64y + 1)(y + 25)`?

Answer 1

`64y^3 + 1601 y^2 + 25y`

Explanation:

Standard form of a polynomial means to write it like this:

`a*y^n + b*y^(n-1) + c*y^(n-2) + cdots + p * y + q`

Where the terms of the polynomial are written in order of decreasing exponents.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In this case, let's start by expanding the two terms `(64y+1)(y+25)`. We can use the FOIL method to do so:

`"FIRST"`

`(color(red)(64y)+1)(color(red)y+25) => color(red)(64y * y) = color(red)(64y^2`

`"OUTER"`

`(color(blue)(64y)+1)(y+color(blue)25) => color(blue)(64y * 25) = color(blue)(1600y`

`"INNER"`

`(64y+color(limegreen)1)(color(limegreen)y+25) => color(limegreen)(1*y) = color(limegreen)(y`

`"LAST"`

`(64y + color(orange)1)(y+color(orange)25) => color(orange)(1*25) = color(orange)(25`

So our polynomial is:

`(64y+1)(y+25) = color(red)(64y^2) + color(blue)(1600y) + color(limegreen)y + color(orange)25 = 64y^2 + 1601y + 25`

Finally, remember that all of this was multiplied by `y` in the original expression:

`y(64y+1)(y+25)`

So, we need to multiply our polynomial by `color(orange)y` to get the final standard form of the polynomial:

`color(orange)y(64y^2+1601y+25) = 64y^2*color(orange)y + 1601y*color(orange)y + 25 * color(orange)y`

`= 64y^3 + 1601 y^2 + 25y`

Final Answer