# 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?

Mar 22, 2016

standard form: $- 3 {k}^{5} - {k}^{2} = 0$
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, $\textcolor{b l u e}{a {x}^{5} + b {x}^{4} + c {x}^{3} + {\mathrm{dx}}^{2} + e x + f = 0}$. Assuming that the given equation is equal to $0$, start by grouping all like terms and simplifying.

$3 {k}^{5} + 4 {k}^{2} - 6 {k}^{5} - 5 {k}^{2} = 0$

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 $\textcolor{red}{\text{term}}$ 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: $3 x , 56 r , 623 u , 6134 a , 23980 f$

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

${\underbrace{- 3 {k}^{5}}}_{\textcolor{red}{\text{term")-underbrace(k^2)_color(red)("term}}} = 0 \textcolor{w h i t e}{X} , \textcolor{w h i t e}{X}$thus:

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \text{number of terms} = 2 \textcolor{w h i t e}{\frac{a}{a}} |}}}$

Determining the Degree
To determine the degree, take the $\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{\text{exponent of each variable}}$ of every $\textcolor{red}{\text{term}}$ in the equation. The highest exponent out of all the terms is the degree of the polynomial.

${\underbrace{- 3 {k}^{5}}}_{\textcolor{red}{\text{term")-underbrace(k^2)_color(red)("term}}} = 0$

• $- 3 {k}^{\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{5}} \textcolor{t e a l}{\to} \textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{5}$

• ${k}^{\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{2}} \textcolor{t e a l}{\to} \textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{2}$

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:

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \text{deg} \left(- 3 {k}^{5} - {k}^{2}\right) = 5 \textcolor{w h i t e}{\frac{a}{a}} |}}}$