Polynomial Written In Standard Form

On this post you volition find the explanation of what a polynomial in standard grade is. You volition also come across examples of polynomials in standard grade and how to put a polynomial in standard form. And finally, you will find solved practice issues on writing a polynomial in standard form.

When is a polynomial in standard class?

A polynomial is in standard form when all its terms are arranged by decreasing order of degree, that is, a polynomial in standard form is a polynomial whose terms are all ordered from highest to everyman degree.

For example, the following polynomial is in standard form:

$x^4+5x^3-4x^2+3x+6$

As you can run into, the previous polynomial is in standard grade considering its terms are bundled past decreasing order. In other words, get-go nosotros take the term ten⁴ which is of fourth degree, secondly there is 5x³ which is of third degree, and then -4x^two which is of 2nd degree, and so 3x which is of start caste and finally half dozen which is the constant term of the polynomial (caste equal to 0).

Therefore, every polynomial whose terms are not ordered in descending lodge it is not in standard grade:

$6x^2+5x+2x^6+4-9x^5$ ❌

Note: there are math books that consider that a polynomial is in standard form when its terms are written in increasing order, such as the following polynomial:

$2-x+6x^2+7x^3$

All the same, the near common is to refer to a polynomial in standard form when its terms are in decreasing gild.

Although the concept of a polynomial in standard form seems to exist a very simple, you should know that this is very of import to perform some polynomial operations. For case, the result of a polynomial division will be wrong if the polynomials are non placed in standard form.

➤ Run into: how to do a long division of polynomials

Examples of polynomials in standard form

Once we have seen the definition of a polynomial in standard form, permit'south encounter several examples of polynomials in standard form to understand better its pregnant.

Example of a polynomial in standard form without constant term:

$x^5+2x^3 + 6x$

Every bit you can come across in the previous example, it is not necessary for a polynomial in standard class to take all the terms of all degrees, as long equally all its terms are ordered from highest to lowest degree it will be considered a polynomial in standard form. Thus, the to a higher place example does not accept a term of caste 4, nor a term of degree 2, nor a constant term, but it is in standard form because all its monomials are ordered in descending order.

Instance of a monic polynomial in standard course:

$x^4+3x^3-5x+7$

Instance of a polynomial in standard form with terms of all degrees:

$3x^6+x^5-6x^4+x^3+2x^2-9x+1$

How to write a polynomial in standard course

To write a polynomial in standard grade, you lot must do the following steps:

Add (or subtract) the like terms of the polynomial.
Write the term with the highest degree showtime.
Write all the other terms in decreasing order of degree.
Call back that a term with a variable only without an exponent is of degree 1.
Remember that a constant term is of degree 0, so it e'er is the last term.

Nosotros are going to rewrite the following polynomial in standard form as an example:

$1-4x^3+x^4+2x$

In this case the polynomial has no like terms, so we don't have to do whatsoever add-on or subtraction.

The highest exponent is 4, thus the first term of the polynomial must be 10⁴.

The next highest caste term is -4xⁱⁱⁱ, so we put it next: x⁴-4x^three.

There is no 2nd caste term, therefore the side by side chemical element is 2x, which is of first degree. And then nosotros have x⁴-4x³+2x.

And the abiding term goes last. So the polynomial expressed in standard course is x^four-4x^three+2x+one.

$x^4-4x^3+2x+1$

Do problems on writing polynomials in standard form

Problem 1

Write the following polynomial in standard class:

$2x^3-5+3x^2+4x^5-3x^6+7x$

There are no like terms in the polynomial, so we must non add or decrease any terms. Therefore, to put the polynomial in standard form we simply take to suit its terms in decreasing order of exponent:

$-3x^6+4x^5+2x^3+3x^2+7x-5$

Trouble 2

Put the following polynomial in standard form:

$3x^2+2+2x-5x^3+5x^2-7x+1$

Beginning of all, we take to add and subtract the like terms of the polynomial:

$(3x^2+5x^2)+(2-1)+(2x-7x)-5x^3$

$8x^2+1-5x-5x^3$

And in one case we have grouped the terms of the same degree, we put them in descending order:

$-5x^3+8x^2-5x+1$

Problem 3

Rewrite the following polynomial with 2 variables in standard form:

$1+4x^2y-y-8xy-x^4y^3+5x^4-y^{10}+x^4y+7x^3y^5$

To notice the degree of a term with two or more variables, you lot take to add together the exponents of all the variables of the term. For case, the term 7x³y^five is of degree 8, since 3+5=8.

Thus, the terms arranged in decreasing social club of degree from left to right (polynomial in standard form) are every bit follows:

$-y^{10}+7x^3y^5-x^4y^3+x^4y+5x^4+4x^2y-8xy-y+1$