# Simplifying Polynomial Expressions

Do you want to know how to simplify polynomial expressions? In this article, you learn how to simplify polynomial expressions easily.

## Step by step guide to solve simplifying polynomial expressions

• In mathematics, a polynomial is an expression consisting of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non–negative integer exponents of variables.
• $$P(x)= a_{n} x^n \ + \ a_{n-1} x^{n \ – \ 1} \ + … + \ a_{2} x^2 \ + \ a_{1} x \ + \ a_{0}$$
• To simplify a polynomial expression, find like terms (terms with same varibale (variables) and same powers. Then combine them.

### Example 1:

Simplify this Polynomial Expression. $$x^2 \ – \ 5 \ x^3 \ + \ 2 \ x^4 \ – \ 4 \ x^3=$$

Solution:

Combine “like” terms: $$− \ 5 x^{3} \ − \ 4 x^{3}= − \ 9x^{3}$$
Then: $$x^2 \ − \ 5 x^{3} \ + \ 2 x^{4} \ − \ 4 x^{3}=x^{2} \ − \ 9 x^{3} \ + \ 2 x^{4}$$
Then write in standard form: $$=2x^{4} \ − \ 9x^{3} \ + \ x^{2}$$

### Example 2:

Simplify this Polynomial Expression. $$(2x^{2} \ – \ x^3 ) \ – \ (x^{3} \ – \ 4 x^{2} )=$$

Solution:

First use distributive property: $$→$$ multiply $$(−)$$ into $$x^{3} \ − \ 4x^{ 2}$$
$$2 x^{2} \ − \ x^{3} \ − \ x^{3} \ + \ 4x^{2}$$
Then combine “like” terms: $$2x^{2} \ − \ x^{3} \ − \ x^{3} \ + \ 4x^{2}=6x^{2} \ − \ 2x^{3}$$
And write in standard form: $$=− \ 2x^{3} \ + \ 6x^{2}$$

### Example 3:

Simplify this Polynomial Expression. $$4x^2-5x^3+15x^4-12x^3=$$

Solution:

Combine “like” terms: $$-5x^3-12x^3= -17x^3$$
Then: $$4x^2-5x^3+15x^4-12x^3=4x^2-17x^3+15x^4$$
Then write in standard form: $$4x^2-17x^3+15x^4=15x^4-17x^3+4x^2$$

### Example 4:

Simplify this expression. $$(2x^2-x^4 )-(4x^4-x^2 )=$$

Solution:

First use distributive property: $$→$$ multiply $$(-)$$ into $$(4x^4-x^2 )$$

$$(2x^2-x^4 )-(4x^4-x^2 )=2x^2-x^4-4x^4+x^2$$
Then combine “like” terms: $$2x^2-x^4-4x^4+x^2=3x^2-5x^4$$
And write in standard form: $$3x^2-5x^4=-5x^4+3x^2$$

## Exercises

### Simplify each polynomial.

• $$\color{blue}{4x^5 – 5x^6 + 15x^5 – 12x^6 + 3 x^6}$$
• $$\color{blue}{(– 3x^5 + 12 – 4x) + (8x^4 + 5x + 5x^5)}$$
• $$\color{blue}{10x^2 – 5x^4 + 14x^3 – 20x^4 + 15x^3 – 8x^4}$$
• $$\color{blue}{– 6x^2 + 5x^2 – 7x^3 + 12 + 22}$$
• $$\color{blue}{12x^5 – 5x^3 + 8x^2 – 8x^5}$$
• $$\color{blue}{5x^3 + 1 + x^2 – 2x – 10x}$$

• $$\color{blue}{– 14x^6 + 19x^5}$$
• $$\color{blue}{2x^5 + 8x^4 + x + 12}$$
• $$\color{blue}{–33x^4 + 29x^3 + 10x^2}$$
• $$\color{blue}{–7x^3 – x^2 + 34}$$
• $$\color{blue}{4x^5 – 5x^3 + 8x^2}$$
• $$\color{blue}{5x^3 + x^2 – 12x + 1}$$