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

Oct 18, 2017

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

Explanation:

Given the expression
$\textcolor{w h i t e}{\text{XXX}} x {y}^{2} + 4 {x}^{4} y + 10 {y}^{2}$

The terms are
$\textcolor{w h i t e}{\text{XXX}} x {y}^{2}$,
$\textcolor{w h i t e}{\text{XXX}} 4 {x}^{4} y$, and
$\textcolor{w h i t e}{\text{XXX}} 10 {x}^{2}$

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
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{4 {x}^{4} y + x {y}^{2} + 10 {x}^{2}}$

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

Since the largest degree of the terms of $4 {x}^{4} y + x {y}^{2} + 10 {x}^{2}$ is $\textcolor{b l u e}{5}$
the polynomial has a degree of $\textcolor{g r e e n}{5}$