How do you put f(x)=x (x-4)^2 (x-8)(x^2+4) in standard form? This is in factored form by the way.

A polynomial is in standard form if it is written as follows:

`P(x) = ax^n + bx^(n-1) + cx^(n-2) +...`

The term with the highest degree comes first and is followed by the other terms in the order of decreasing powers of the variable.

To express the function

`f(x) = x(x-4)^2(x-8)(x^2+4)`

in standard form, let's first expand repeated factor.

The expanded form of the repeated factor is:

`@` `(x-4)^2=(x-4)(x-4) = x^2-4x-4x+16=x^2-8x+16`

The function becomes:

`f(x)=x(x^2-8x+16)(x-8)(x^2+4)`

...

A polynomial is in standard form if it is written as follows:

`P(x) = ax^n + bx^(n-1) + cx^(n-2) +...`

The term with the highest degree comes first and is followed by the other terms in the order of decreasing powers of the variable.

To express the function

`f(x) = x(x-4)^2(x-8)(x^2+4)`

in standard form, let's first expand repeated factor.

The expanded form of the repeated factor is:

`@` `(x-4)^2=(x-4)(x-4) = x^2-4x-4x+16=x^2-8x+16`

The function becomes:

`f(x)=x(x^2-8x+16)(x-8)(x^2+4)`

Then, multiply the factors. Let's start with the factors at the left.

`@` `x(x^2-8x+16)=x^3-8x^2+16x`

The function transforms to three factors.

`f(x)= (x^3-8x^2+16x)(x-8)(x^2+4)`

Then, multiply (x^3-8x^2+16x) with (x-8).

`@`  `(x^3-8x^2+16x)(x-8)`

   `=x^4-8x^3 -8x^3+64x^2+16x^2-128x`

  `= x^4-16x^3+80x^2-128x`

f(x) is reduced to two factors.

`f(x) = (x^4-16x^3+80x^2-128x)(x^2+4)`

Multiply these two factors.

`@`  `(x^4-16x^3+80x^2-128x)(x^2+4)`
  `= x^6 +4x^4-16x^5-64x^3+80x^4+320x^2-128x^3-512x`
  `= x^6 -16x^5+84x^4-192x^3+320x^2-512x`

The function is now converted to standard form.

Therefore the standard form of

`f(x) = x(x-4)^2(x-8)(x^2+4) `

is

`f(x)= x^6 -16x^5+84x^4-192x^3+320x^2-512x` .