How to Simplify Polynomials

How to Simplify Polynomials

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As the name suggests, the word “polynomial” can be cut into two words “poly” and “nomial” which means multiple expressions. Therefore, a polynomial refers to an algebraic or a mathematical expression that consists of one or more terms. 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 \(x\).

Degree of a Polynomial Expression

To know the degree of a single variable expression, follow these steps:

  • First, simplify the polynomial expression.
  • Then, write the polynomial expression in the standard form.
  • Next, you should check and select the highest exponent of the variable term.

Ex: \(4x^3 \ + \ 3x^2 \ + \ 7\) has the degree \(3\) since the highest exponent of \(x\) is \(3\).

To know the degree of a multivariable polynomial expression, follow these steps:

  • First, simplify the polynomial expression.
  • Then, write the polynomial expression in the standard form.
  • Next, you should check and select the highest exponent of the variable term. Note: you should add the powers of different variables too.

Ex: \(x^3 \ + \ 3x^2y^3 \ + \ 4\) has the degree \(5\), since the middle term has the power of \(x \ = \ 2\) and \(y \ = \ 3\), so the addition comes to \(5\).

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 Polynomials

1) \(16 \ + \ 3x^3 \ - \ 7x^2 \ - \ 2 = \)

2) \(25 \ + \ 7x^3 \ - \ 5x^2 \ - \ 4 = \)

3) \(18 \ + \ 3(-2x^3 \ - \ 4x^2) \ - \ 2 + x = \)

4) \(3x (x \ + \ 5x^2 \ - \ 2x^4) = \)

5) \(18x (x \ + \ 6x^2 \ - \ 7x^4) = \)

6) \((-x \ + \ 7x^2)x = \)

7) \((x \ - \ 20x^2)(x \ + \ 2) = \)

8) \((-x \ + \ 14x^2)x = \)

9) \((x \ - \ 5x^2)(x \ + \ 3) = \)

10) \((x \ - \ 6x^2)(x \ + \ 3) = \)

 
1) \(16 \ + \ 3x^3 \ - \ 7x^2 \ - \ 2 = \)\( \ \color{red}{3x^3 \ - \ 7x^2 \ + \ 14}\)
2) \(25 \ + \ 7x^3 \ - \ 5x^2 \ - \ 4 = \)\( \ \color{red}{7x^3 \ - \ 5x^2 \ + \ 21}\)
3) \(18 \ + \ 3(-2x^3 \ - \ 4x^2) \ - \ 2 + x = \)\( \ \color{red}{-6x^3 \ - \ 12x^2 \ + \ x \ + \ 16}\)
4) \(3x (x \ + \ 5x^2 \ - \ 2x^4) = \)\( \ \color{red}{-6x^4 \ + \ 15x^3 \ + \ 3x^2}\)
5) \(18x (x \ + \ 6x^2 \ - \ 7x^4) = \)\( \ \color{red}{-126x^4 \ + \ 108x^3 \ + \ 18x^2}\)
6) \((-x \ + \ 7x^2)x = \)\( \ \color{red}{7x^3 \ - \ x^2}\)
7) \((x \ - \ 20x^2)(x \ + \ 2) = \)\( \ \color{red}{-20x^3 \ - \ 39x^2 \ + \ 2x}\)
8) \((-x \ + \ 14x^2)x = \)\( \ \color{red}{14x^3 \ - \ x^2}\)
9) \((x \ - \ 5x^2)(x \ + \ 3) = \)\( \ \color{red}{-5x^3 \ - \ 14x^2 \ + \ 3x}\)
10) \((x \ - \ 6x^2)(x \ + \ 3) = \)\( \ \color{red}{-6x^3 \ - \ 17x^2 \ + \ 3x}\)

Simplifying Polynomials Practice Quiz