## How to simplify Polynomial Expressions

### Polynomial- Definition!

As the name suggests, the word “polynomial” can be cut into two words “poly” and “nomial” which means multiple expressions. This a polynomial means an algebraic or a mathematical expression that contains at least two terms with a math operation between them. Also, the expressions in the polynomial terms could be a constant, a variable, or even a mixed term with a co-efficient. Also, in a polynomial, the power of the variable (let’s suppose $$x$$) should be an integer and not a fractional power like a squared root or a cube root.
For example, $$2x^\frac{1}{3} \ + \ 3x \ + \ 1$$ is not a polynomial as the first term has a fraction power of $$x$$.

### 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.
Ex: $$6x^2, \ 7x^3, \ 67x$$, etc.
Binomial: Binomials consist of just two terms and they must have a mathematical operation in between.
Ex: $$6x^2 \ + \ 7x^3, \ 8x^2 \ + \ 7y, \ xy^2 \ + \ 3x$$, etc.
Trinomial: Trinomial consist of three terms and between every two terms there is a mathematical operation. Ex: $$2x^2 \ + \ 3x \ + \ 10, \ 4x^3 \ + \ 3x^2 \ + \ 7, \ 3x^2y \ + \ 4xy \ + \ 6$$, etc.

### Simplifying Polynomial Expressions

To simplify a polynomial expression, apply the below-mentioned steps:

• First, simplify the expression by adding/subtracting the like terms.
• Also, wherever possible, use the distributive property.

Some Examples:

• $$4x^3 \ + \ 3x^3 \ + \ 2x^2 \ - \ x^2 \ + \ 9 \ = \ 7x^3 \ + \ x^2 \ + \ 9$$.
• $$2x^3 \ - \ 5x^3 \ + \ 7x^2 \ - \ x^2 \ + \ 5 \ = \ -3x^3 \ + \ 6x^2 \ + \ 5$$.
• $$-x^3 \ + \ 7x^3 \ - \ x^2 \ + \ x^2 \ + \ 7 \ = \ 6x^3 \ + \ 7$$.
• $$12x^3 \ + \ 15x^3 \ - \ 2x^2 \ - 8x^2 \ + \ 19 \ = \ 27x^3 \ - \ 10x^2 \ + \ 19$$.

### Exercises for Simplifying Polynomial Expressions

1) $$4x + 7x^2 - x^3 + 5x =$$

2) $$7x^3 + 6x^2 - 6x^3$$ + $$13x^2 - 13x^3 + 12x^2 =$$

3) $$18x + 2x^2 - x^3 + 5x =$$

4) $$11x + 3x^2 - x^3 + 5x =$$

5) $$20x + 3x^2 + x^3 + 4x - 4 =$$

6) $$13x + 7x^2 + x^3 + 6x - 6 =$$

7) $$(16x^3 + 7x^2 - 7x^3)$$ + $$(23x^2 - 23x^3 + 14x^2) =$$

8) $$(2x^3 + 6x^2 - 7x^3)$$ + $$(9x^2 - 8x^3 + 13x^2) =$$

9) $$(9x^3 + 2x^2 - 7x^3)$$ + $$(16x^2 - 11x^3 + 9x^2) =$$

10) $$(10x^3 + 5x^2 - 5x^3)$$ + $$(15x^2 - 15x^3 + 10x^2) =$$

1) $$4x + 7x^2 - x^3 + 5x =$$$$\ \color{red}{-x^3 + 7x^2 + 9x}$$
2) $$7x^3 + 6x^2 - 6x^3$$ + $$13x^2 - 13x^3 + 12x^2 =$$$$\ \color{red}{-12x^3 +31x^2}$$
3) $$18x + 2x^2 - x^3 + 5x =$$$$\ \color{red}{-x^3 + 2x^2 + 23x}$$
4) $$11x + 3x^2 - x^3 + 5x =$$$$\ \color{red}{-x^3 + 3x^2 + 16x}$$
5) $$20x + 3x^2 + x^3 + 4x - 4 =$$$$\ \color{red}{x^3 + 3x^2 + 24x - 4}$$
6) $$13x + 7x^2 + x^3 + 6x - 6 =$$$$\ \color{red}{x^3 + 7x^2 + 19x - 6}$$
7) $$(16x^3 + 7x^2 - 7x^3)$$ + $$(23x^2 - 23x^3 + 14x^2) =$$$$\ \color{red}{-14x^3 +44x^2}$$
8) $$(2x^3 + 6x^2 - 7x^3)$$ + $$(9x^2 - 8x^3 + 13x^2) =$$$$\ \color{red}{-13x^3 +28x^2}$$
9) $$(9x^3 + 2x^2 - 7x^3)$$ + $$(16x^2 - 11x^3 + 9x^2) =$$$$\ \color{red}{-9x^3 +27x^2}$$
10) $$(10x^3 + 5x^2 - 5x^3)$$ + $$(15x^2 - 15x^3 + 10x^2) =$$$$\ \color{red}{-10x^3 +30x^2}$$

## Simplifying Polynomial Expressions Practice Quiz

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