Question

How do you express `(x-5)^2` in standard form?

Answer 1

`(x-5)^2 = x^2-10x+25`

Explanation:

The standard form of a polynomial in one variable is a sum of terms in decreasing order of degree.

For brevity, terms with negative coefficients are usually written as subtractions rather than adding the additive inverses. That is, we would write `x^2-3x+2` rather than `x^2+(-3)x+2`

In order to express `(x-5)^2` in standard form, we need to multiply it out, combining terms of like degree. We can use the FOIL mnemonic to help:

`(x-5)^2 = (x-5)(x-5)`

`color(white)((x-5)^2) = overbrace((x)(x))^"First" + overbrace((x)(-5))^"Outside" + overbrace((-5)(x))^"Inside" + overbrace((-5)(-5))^"Last"`

`color(white)((x-5)^2) = x^2-5x-5x+25`

`color(white)((x-5)^2) = x^2-10x+25`

Alternatively, we can recognise a pattern and use it.

For example:

`(a+b)^2 = a^2+2ab+b^2`

`(a-b)^2 = a^2-2ab+b^2`

So we could take the second of these and put `a=x`, `b=5` to find:

`(x-5)^2 = x^2-2(x)(5)+5^2`

`color(white)((x-5)^2) = x^2-10x+25`