How to Simplify Polynomial Expressions? (+FREE Worksheet!)

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Related Topics

• How to Translate Phrases into an Algebraic Statement
• How to Simplify Variable Expressions
• How to Use the Distributive Property
• How to Evaluate One Variable
• How to Evaluate Two Variables

Step-by-step guide to 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 the same variable (variables) and same powers. Then combine them.

Simplifying Polynomial Expressions For education statistics and research National Center for Education Statistics.

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Simplifying Polynomial Expressions – Example 1:

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

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 \(x^{3} \ − \ 4x^{ 2}\)
\((2x^{2} \ – \ x^3 ) \ – \ (x^{3} \ – \ 4 x^{2} )=\) \(2 x^{2} \ − \ x^{3} \ − \ x^{3} \ + \ 4x^{2} \)
Combine “like” terms: \(2x^{2} + \ 4x^{2}=6x^{2} \, − x^{3} \ − \ x^{3} \ = − \ 2x^{3}\)

Then \(2x^{2} \ − \ x^{3} \ − \ x^{3} \ + \ 4x^{2}=6x^{2} \ − \ 2x^{3}\)
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: \(=\) \( 15x^4-17x^3+4x^2\)

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Simplifying Polynomial Expressions – Example 4:

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

Solution:

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

\( (2x^2-x^4 )-(4x^4-x^2 )=2x^2-x^4-4x^4+x^2 \)
Combine “like” terms: \(2x^2+x^2=3x^2\), \(-x^4-4x^4=-5x^4\)

Then: \(2x^2-x^4-4x^4+x^2=3x^2-5x^4\)
And write in standard form: \(=\) \(-5x^4+3x^2\)

Exercises for Simplifying Polynomial Expressions

Simplifying Polynomial Expressions

Simplify each polynomial.

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

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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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