## How to Write Polynomials in Standard Form

### What is a Polynomial?

A polynomial can be broken down into two words: "poly" and "nomial," which both signify many  terms, as suggested by the name of the concept. In mathematics, a polynomial is an algebraic or mathematical expression that has at least two terms that are linked together by a mathematical operation. It should be noted that expressions within the polynomial terms could be a constant, a variable, or even a mixed term that contains a co-efficient. A polynomial should also have the power of the variable (let's say $$x$$) be an integer rather than a fractional power such as the squared root or the cube root.
For example, $$7x^{\frac{1}{2}} \ - \ 5x \ + \ 13$$ is not a polynomial as the first term has a fraction power of $$x$$.

### Different Types of Polynomials

Based on the number of terms, there are 3 different types of polynomials. They are monomials, binomials and trinomials. Let’s discuss more about them:

• Monomial: They consist of just a single term but there is one condition. The term cannot be zero. Also, monomials can all be mixed terms as well as just pure variable terms. Ex: $$5x^2, \ 6x^3, \ 69x$$, etc.
• Binomial: Binomials consist of just two terms and they must have a mathematical operation in between. Binomials can have mixed terms, pure variable terms, or even pure constant terms in the expression. Ex: $$2x^2 \ + \ 5x^3, \ 3x^2 \ + \ 8y, \ xy^2 \ + \ 2x$$, etc.
• Trinomial: Trinomial consist of three terms and between every two terms there is a mathematical operation. Trinomials too can have mixed terms, pure variable terms, and even pure constant terms in the expression. Ex: $$2x^2 \ + \ 3x \ + \ 10, \ 4x^3 \ + \ 3x^2 \ + \ 7, \ 3x^2y \ + \ 4xy \ + \ 6$$, etc.

### Standard Form of Polynomials

The standard form of a polynomial function $$f(x)$$ of the degree $$n$$ is:
$$F(x) \ = \ a_{n}x^n \ + \ a_{n-1}x^{n \ - \ 1} \ + \ …… \ + \ a_{1}x \ + \ a_{0}$$, where the first term is the term with the biggest power of the variable (in this case $$x$$).
To write a polynomial in the standard form, we must arrange all powers of $$x$$ in descending order. For example, in $$4x^2 \ - \ 9x^3 \ + \ 13x \ - \ 7$$, we write the standard form as:
$$-9x^3 \ + \ 4x^2 \ + \ 13x \ – \ 7$$.

### Exercises for Writing Polynomials in Standard Form

1) $$-20x^2 \ - \ x \ + \ 24x^3 =$$

2) $$x \ -\ 2(9x^3 \ - \ x) \ - \ 13x^4 =$$

3) $$x \ -\ 2(16x^3 \ - \ x) \ - \ 20x^4 =$$

4) $$-3x^2 \ - \ 3 \ + \ 6x^3 =$$

5) $$-18x^2 \ - \ 7 \ + \ 19x^3 =$$

6) $$-2(7x^3 \ - \ x) \ - \ 9x^4 =$$

7) $$16x^2 - 18x^3 =$$

8) $$-2(22x^3 \ - \ x) \ - \ 24x^4 =$$

9) $$-5x^2 \ - \ x \ + \ 9x^3 =$$

10) $$-6x^2 \ - \ x \ + \ 10x^3 =$$

1) $$-20x^2 \ - \ x \ + \ 24x^3 =$$$$\ \color{red}{24x^3 - 20x^2 \ - \ x}$$
2) $$x \ -\ 2(9x^3 \ - \ x) \ - \ 13x^4 =$$$$\ \color{red}{-13x^4 - 18x^3 \ + \ 3x}$$
3) $$x \ -\ 2(16x^3 \ - \ x) \ - \ 20x^4 =$$$$\ \color{red}{-20x^4 - 32x^3 \ + \ 3x}$$
4) $$-3x^2 \ - \ 3 \ + \ 6x^3 =$$$$\ \color{red}{6x^3 - 3x^2 \ - \ 3}$$
5) $$-18x^2 \ - \ 7 \ + \ 19x^3 =$$$$\ \color{red}{19x^3 - 18x^2 \ - \ 7}$$
6) $$-2(7x^3 \ - \ x) \ - \ 9x^4 =$$$$\ \color{red}{-9x^4 - 14x^3 \ + \ 2x}$$
7) $$16x^2 - 18x^3 =$$$$\ \color{red}{-18x^3 + 16x^2 }$$
8) $$-2(22x^3 \ - \ x) \ - \ 24x^4 =$$$$\ \color{red}{-24x^4 - 44x^3 \ + \ 2x}$$
9) $$-5x^2 \ - \ x \ + \ 9x^3 =$$$$\ \color{red}{9x^3 - 5x^2 \ - \ x}$$
10) $$-6x^2 \ - \ x \ + \ 10x^3 =$$$$\ \color{red}{10x^3 - 6x^2 \ - \ x}$$

## Writing Polynomials in Standard Form Practice Quiz

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