How to Simplify Polynomial Expressions

How to Simplify Polynomial Expressions

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

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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.

Simplifying Polynomial Expressions – 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: \(=2x^{4} \ − \ 9x^{3} \ + \ x^{2} \)

Simplifying Polynomial Expressions – 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: \(=− \ 2x^{3} \ + \ 6x^{2}\)

Simplifying Polynomial Expressions – 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: \(4x^2-17x^3+15x^4=15x^4-17x^3+4x^2\)

Simplifying Polynomial Expressions – 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 for Simplifying Polynomial Expressions

Simplify each polynomial.

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

Download Simplifying Polynomial Expressions Worksheet

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Answers

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

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