# Polynomial Equations

Polynomials are one of the significant concepts of Mathematics, and so are Polynomial Equations, where the relation between numbers and variables are explained in a pattern.

In Math, there are a variety of equations formed with algebraic expressions. Polynomial Equations are also a form of algebraic equations.

This mini-lesson will give an overview of polynomial equation definition, polynomial formula, the difference between polynomial and equation, polynomial equation formula & polynomial equation examples.

Let's see what Henry's teacher had to ask him.

She asked 'Whether the given equation was a polynomial equation or not?'

Stay tuned with Henry to learn more about polynomial equations!!

**Lesson Plan **

**Introduction**

In algebra, almost all equations are polynomial equations.

Consider the below equation:

\( 2x^{2} + 3x + 1 = 0\)

Here, \(2x^{2} + 3x + 1\) is basically a polynomial expression which has been set equal to zero, thus forming a polynomial equation.

Now, let's explore more details about polynomial equations.

**What Is A Polynomial Equation?**

**Polynomial Equation Definition**

An equation formed with variables, exponents, and coefficients together with operations and an equal sign is called a polynomial equation.

It has different exponents. The higher one gives the degree of the equation.

Usually, the polynomial equation is expressed in the form of \(\mathrm{a}_{\mathrm{n}}\left(\mathrm{x}^{\mathrm{n}}\right)\).

- a is the coefficient
- x is the variable
- n is the exponent.

On putting the values of a and n, we get a polynomial function of degree n.

\(F(x)= \mathrm{a}_{\mathrm{n}}\left(\mathrm{x}^{\mathrm{n}}\right) = a_{n} x^{n}+....+a_{0}=0\)

For any polynomial P(x), the polynomial equation will be P(x) = 0

**Polynomial Equation Examples**

Examples |
Polynomial Equation or not |
Reason |

\(x^2 + 3\sqrt{x} + 1 = 0\) | No | fractional power of \(x\) i.e; \(\frac{1}{2}\) which is a non integer value |

\(x^2 + \sqrt{3}x + 1= 0\) | Yes | the constant has a fractional power, not the variable |

**Polynomial Equation Formula**

**General Polynomial Equation Formula**

\(\mathrm{F}(\mathrm{x})=\mathrm{a}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}}+ . .+\mathrm{rx}+\mathrm{s}\)

- n is a natural number

\(a^{n}-b^{n}=(a-b)\left(a^{n-1}+a^{n-2} b+\ldots\right)\)

- n is even number

\(a^{n}+b^{n}=(a+b)\left(a^{n-1}-a^{n-2} b+\ldots\right)\)

- n is odd number

\(a^{n}+b^{n}=(a+b)\left(a^{n-1}-a^{n-2} b+\ldots\right)\)

- Determine a few characteristics of an algebraic equation to not to be considered as a polynomial equation.

**Types of Polynomial Equations**

**1. Monomial Equations/Linear Equations**

Equations with one variable term.

\(ax + b = 0\) |

**2. Binomial Equations/Quadratic Equations**

Equations with two variable terms.
\(a x^{2}+b x+c=0\) |

**3. Trinomial Equations/Cubic Equations:**

Equations with three variable terms.
\(a x^{3}+b x^{2}+c x+d=0\) |

- The degree of a polynomial equation is the highest power of the variable in the equation.
- Solving an equation is finding those values of the variables which satisfy the equation. You can learn more about it on the solving polynomials page.
- You can also find a polynomial equation when roots are known.

**What Is the Difference between Polynomial and Equation? **

A polynomial is the parent term used to describe a certain type of algebraic expressions that contain variables, constants, and involve the operations of addition, subtraction, multiplication, and division along with only positive powers associated with the variables.

Example: 2x + 3

An equation is a mathematical statement with an 'equal to' symbol between two algebraic expressions that have equal values.

Example: 2x + 3 = 7

**Solved Examples**

Example 1 |

Help Justin classify whether the equations given below are polynomial equations or not.

- \(\sqrt{x^2+y^2} + 2=0\)
- \(x^2 + 3x + 2=0\)
- \(\frac{x}{2} + 3x^2 + 5=0\)
- \(3x^3 - \sqrt{2}x + 1=0\)
- \(\frac{2}{x+3}=0\)

**Solution**

Justin will check for two conditions in the given equations.

- If the equation has a non-integer exponent of the variable.
- If the equation has any variable in the denominator.

If an equation has the above-mentioned features, it will not be a polynomial equation.

Equations | Criteria to be checked | Polynomial or not |
---|---|---|

\(\sqrt{x^2+y^2} =0\) | The equation has a non-integer exponent. \(\frac{1}{2}\) is the exponent of variable here inside a radical. | No |

\(x^2 + 2=0\) | - | Yes |

\(\frac{x}{2} + 3x^2 =0\) | - | Yes |

\(3x^3 - \sqrt{2}x =0\) | - | Yes |

\(\frac{2}{x+3}=0\) | The equation consists of operations other than addition, subtraction, multiplication, and division by constants as here 2 is being divided by a variable. | No |

Example 2 |

Rustin comprehended a math problem and formed an equation \(2x^4 - 5x^3 + 9x^3 - 3x^4 = 0\). Help him to simplify it further and classify it as a monomial, binomial or trinomial equation!

**Solution**

Simplification of equation:

\(2x^4 - 5x^3 + 9x^3 - 3x^4 = 4x^3 - x^4 \).

Which gives, \(4x^3 - x^4= 0 \).

The obtained output has two terms which mean it is a binomial equation.

\(\therefore\) It's a binomial equation. |

Example 3 |

Maria came across a problem according to which \((2x+6)\) and \((x-8)\) are the roots of the polynomial equation.

Can you find the polynomial equation when its respective roots are given?

Justify your answer.

**Solution**

Roots of equation = \((2x+6)\) and \((x-8)\)

For any polynomial P(x), the polynomial equation will be P(x) = 0

This mean (2x+6) (x-8) = 0 will give the polynomial equation.

Let's solve it.

\(2x(x-8) + 6(x-8) = 0\)

\(2x^{3} - 16x + 6x - 48 = 0\)

\(2x^{3} - 10x - 48 = 0\)

Hence, the polynomial equation is \(2x^{3} - 10x - 48 = 0\)

\(\therefore\) \(2x^{3} - 10x - 48 = 0\) |

**Interactive Questions **

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize **

We hope you enjoyed learning about Polynomial Equations with the simulations and practice questions. Now you will be able to easily solve problems on Polynomial Equations in math with multiple math examples you learned today.

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**Frequently Asked Questions (FAQs)**

### 1. How will you know if an equation is a polynomial equation?

It is basically a set of polynomial expressions equated to 0.

### 2. What is the difference between a polynomial and an equation?

A polynomial is a certain type of algebraic expressions that contain variables, constants, and involve the operations of addition, subtraction, multiplication, and division along with only powers associated with the variables.

An equation is a mathematical statement with an 'equal to' symbol between two algebraic expressions that have equal values.

### 3. What are the different types of polynomial equations?

The different types of polynomial equations are- monomial equations, binomial equations, trinomial equations, and quadratic equations.

### 4. What is a polynomial formula?

A polynomial formula is a polynomial function.

\(\mathrm{F}(\mathrm{x})=\mathrm{a}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}}+\ldots \ldots . .+\mathrm{rx}+\mathrm{s}\)

### 5. What is not a polynomial equation?

Any algebraic equation with a negative exponent or fractional exponent is not a polynomial equation.

### 6. What is the general form for a polynomial equation?

The general form of polynomial equation: \(F(x)=a_{n} x^{n}+.......+a_{1} x+a_{0}=0\)

### 7. How do you solve polynomial equations?

The polynomial equations can be solved by factoring them in terms of degree and variables present in the equation.

### 8. How to find the degree of polynomial equations?

The highest power of the variable term in the polynomial is the degree of the polynomial.

### 9. How do you find the roots of a polynomial equation?

The roots of a polynomial are the values of x, at which the function equals zero, and are thus called zeros of the polynomial. Roots of a polynomial can be calculated by factoring as well as graphing.