Question

How do you write a polynomial in standard form, then classify it by degree and number of terms `xy^2 + 4x^4y + 10x^2`?

Answer 1

In standard form: `4x^4y+xy^2+10x^2`
The polynomial is of degree `5`
and has `3` terms

Explanation:

Given the expression
`color(white)("XXX")xy^2+4x^4y+10y^2`

The terms are
`color(white)("XXX")xy^2`,
`color(white)("XXX")4x^4y`, and
`color(white)("XXX")10x^2`

The degree of a term is the sum of the exponents of all variables in the term
`color(white)("XXX"){: (ul("term"),color(white)("xx"),ul("degree of term")), (xy^2=x^color(red)1y^color(red)2,,=color(red)1+color(red)2=color(blue)3), (4x^4y=4x^color(red)4y^color(red)1,,=color(red)4+color(red)1=color(blue)5), (10x^2=10x^color(red)2,,=color(red)2=color(blue)2) :}`

A polynomial is in standard form if its terms are arranged in descending value of their degrees.

So the standard form for the given polynomial would be
`color(white)("XXX")color(green)(4x^4y+xy^2+10x^2)`

The degree of a polynomial is the largest degree of any of its terms.

Since the largest degree of the terms of `4x^4y+xy^2+10x^2` is `color(blue)5`
the polynomial has a degree of `color(green)5`