Question

How do you write a polynomial in standard form, then classify it by degree and number of terms `14x^2 - 8 + 5x - 6x^2 + 2x`?

Answer 1

standard form: `8x^2+7x-8=0`
number of terms: `3`
degree of polynomial in standard form: `2`

Explanation:

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

`14x^2-8+5x-6x^2+2x=0`

`14x^2-6x^2+5x+2x-8=0`

#color(green)(|bar(ul(color(white)(a/a)8x^2+7x-8=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:

`underbrace(8x^2)_color(red)("term")+underbrace(7x)_color(red)("term")-underbrace(8)_color(red)("term")=0color(white)(X),color(white)(X)`thus:

`color(green)(|bar(ul(color(white)(a/a)"number of terms"=3color(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(8x^2)_color(red)("term")+underbrace(7x)_color(red)("term")-underbrace(8)_color(red)("term")=0`

• `8x^color(darkorange)2color(teal)(->)color(darkorange)2`

• `7x^color(darkorange)1color(teal)(->)color(darkorange)1`

• `8color(teal)(->)0`

As you can see, the highest exponent is `2`, so the degree of the polynomial is `2`. The degree of the polynomial in standard form is written mathematically as:

`color(green)(|bar(ul(color(white)(a/a)"deg"(8x^2+7x-8)=2color(white)(a/a)|)))`