Question

How do you write a polynomial in standard form, then classify it by degree and number of terms `7x + 2 - 4x^2 + 8x^3`?

Answer 1

standard form: `8x^3-4x^2+7x+2`
degree: `3`
number of terms: `4`

Explanation:

A polynomial in standard form is the sum/difference of terms arranged in descending degree order.

The degree of a term is the exponent of the variable of the term (if there are multiple variables in the term it is the sum of the exponents).

`{: (color(black)("term"),color(white)("XXXX"),color(black)("degree")), (7x,,1), (2,,0), (-4x^2,,2), (8x^3,,3) :}`

The degree of a polynomial is the degree of the term with the largest degree.

The number of terms is the number of components added/subtracted when the polynomial is expressed in standard form.

This last point is trivial in this case but note, for example,
`color(white)("XXX")(2x+3)(5x+1)` has `underline(3)` terms
since when expressed in standard form:
`color(white)("XXX")(2x+3)(5x+1)=underbrace(10x^2)+underbrace(17x)+underbrace(3)`
`color(white)("XXXXXXXXXXXXXXX")1color(white)("XXXX")2color(white)("XXX")3`