Question

How do you write a polynomial in standard form, then classify it by degree and number of terms `3k^5+4k^2-6k^5-5k^2`?

Answer 1

standard form: `-3k^5-k^2=0`
number of terms: `2`
degree of polynomial in standard form: `5`

Explanation:

Determining the Standard Form of the Polynomial
Notice how your equation follows the general equation of a quintic function in standard form, `color(blue)(ax^5+bx^4+cx^3+dx^2+ex+f=0)`. Assuming that the given equation is equal to `0`, start by grouping all like terms and simplifying.

`3k^5+4k^2-6k^5-5k^2=0`

#color(green)(|bar(ul(color(white)(a/a)-3k^5-k^2=0color(white)(a/a) |)))#

Determining the Number of Terms
The number of terms of the polynomial in standard form can be found by first defining what a term is. A `color(red)("term")` is a single number, a variable, or a number and variable multiplied together.

For example (but not limited to):

• Single numbers: `3, 56, 623, 6134, 23980`
• Variables: `x, r, u, a, f`
• Number and variable: `3x, 56r, 623u, 6134a, 23980f`

Going back to your equation, the terms would be the following, where the positive and negative signs are ignored after the first term:

`underbrace(-3k^5)_color(red)("term")-underbrace(k^2)_color(red)("term")=0color(white)(X),color(white)(X)`thus:

`color(green)(|bar(ul(color(white)(a/a)"number of terms"=2color(white)(a/a)|)))`

Determining the Degree
To determine the degree, take the `color(darkorange)("exponent of each variable")` of every `color(red)("term")` in the equation. The highest exponent out of all the terms is the degree of the polynomial.

`underbrace(-3k^5)_color(red)("term")-underbrace(k^2)_color(red)("term")=0`

• `-3k^color(darkorange)5color(teal)(->)color(darkorange)5`

• `k^color(darkorange)2color(teal)(->)color(darkorange)2`

As you can see, the highest exponent is `5`, so the degree of the polynomial is `5`. The degree of the polynomial in standard form is written mathematically as:

`color(green)(|bar(ul(color(white)(a/a)"deg"(-3k^5-k^2)=5color(white)(a/a)|)))`