How to simplify Polynomial Expressions

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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:

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

Free printable Worksheets

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

Answers

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) \((10x^3 + 5x^2 - 5x^3) \) + \( (15x^2 - 15x^3 + 10x^2) = \)\( \ \color{red}{-10x^3 +30x^2} \)

How to simplify Polynomial Expressions