[Math] Is the simplest form of a quadratic equation factored form or standard form

Question

I've done a bit of research about what defines simplest form, but I could not find a clear answer. Suppose we had to choose:

$$(x – 4)(x + 2) \quad\text{or}\quad x^2 – 2x – 8$$

A question asked me, "Explain how you know if a quadratic equation is in simplest form." Is it the factored form or the standard form that is considered "simplest form"?

Answer 1

Short answer: there is no One True Meaning to the phrase "simplest form"; you'll have to infer from context, or consult your textbook / professor to see if what they want is explicitly stated.

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Long answer: it depends on what you're trying to do. For example:

• The first form is clearly simpler if the things you're interested in are things like the roots of the polynomial, or multiplying polynomials.
• The second form is clearly simpler if the things you're interested in are things like the coefficients of your polynomial, or adding polynomials.

Sometimes, completely different things are the simplest form; e.g. sometimes the most useful way to represent a polynomial is by a list of values at various points:

$f(0) = -8$ and $f(1) = -9$ and $f(-1) = -5$

or sometimes you want to write it as a power series about some value other than 0:

$(x+1)^2 - 4(x+1) - 5$

For quadratic polynomials in particular, it is sometimes considered simpler to express them after completing the square

$(x-1)^2 - 9$

or even to write it as a sum or difference of squares:

$(x-1)^2 - 3^2$

Regarding factored form, in some contexts -- e.g. you're mainly doing integer and rational number arithmetic, you would consider

$x^2 - 2$

to already be in factored form. But in other contexts,

$(x - \sqrt{2}) (x + \sqrt{2})$

is the correct factored form.