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

4 x + 7 x^{2} - x^{3} + 5 x =

$4x + 7x^2 - x^3 + 5x =$

- x^{3} + 7 x^{2} + 9 x

$\ \color{red}{-x^3 + 7x^2 + 9x}$

7 x^{3} + 6 x^{2} - 6 x^{3}

$7x^3 + 6x^2 - 6x^3$ +

13 x^{2} - 13 x^{3} + 12 x^{2} =

$13x^2 - 13x^3 + 12x^2 =$

- 12 x^{3} + 31 x^{2}

$\ \color{red}{-12x^3 +31x^2}$

18 x + 2 x^{2} - x^{3} + 5 x =

$18x + 2x^2 - x^3 + 5x =$

- x^{3} + 2 x^{2} + 23 x

$\ \color{red}{-x^3 + 2x^2 + 23x}$

11 x + 3 x^{2} - x^{3} + 5 x =

$11x + 3x^2 - x^3 + 5x =$

- x^{3} + 3 x^{2} + 16 x

$\ \color{red}{-x^3 + 3x^2 + 16x}$

20 x + 3 x^{2} + x^{3} + 4 x - 4 =

$20x + 3x^2 + x^3 + 4x - 4 =$

x^{3} + 3 x^{2} + 24 x - 4

$\ \color{red}{x^3 + 3x^2 + 24x - 4}$

13 x + 7 x^{2} + x^{3} + 6 x - 6 =

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

x^{3} + 7 x^{2} + 19 x - 6

$\ \color{red}{x^3 + 7x^2 + 19x - 6}$

(16 x^{3} + 7 x^{2} - 7 x^{3})

$(16x^3 + 7x^2 - 7x^3)$ +

(23 x^{2} - 23 x^{3} + 14 x^{2}) =

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

- 14 x^{3} + 44 x^{2}

$\ \color{red}{-14x^3 +44x^2}$

(2 x^{3} + 6 x^{2} - 7 x^{3})

$(2x^3 + 6x^2 - 7x^3)$ +

(9 x^{2} - 8 x^{3} + 13 x^{2}) =

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

- 13 x^{3} + 28 x^{2}

$\ \color{red}{-13x^3 +28x^2}$

(9 x^{3} + 2 x^{2} - 7 x^{3})

$(9x^3 + 2x^2 - 7x^3)$ +

(16 x^{2} - 11 x^{3} + 9 x^{2}) =

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

- 9 x^{3} + 27 x^{2}

$\ \color{red}{-9x^3 +27x^2}$

10)

(10 x^{3} + 5 x^{2} - 5 x^{3})

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

(15 x^{2} - 15 x^{3} + 10 x^{2}) =

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

- 10 x^{3} + 30 x^{2}

$\ \color{red}{-10x^3 +30x^2}$

How to simplify Polynomial Expressions