Polynomial Equations

A polynomial equation is an equation with a polynomial expression equals to zero. It is made up of variables, numbers, and basic operations like addition, subtraction, and multiplication. The highest power of the variable shows the degree of the polynomial. In a polynomial equation, the exponents must be nonnegative numbers; negative numbers or fractions are not     allowed. If this condition is satisfied, the equation can be classified as a polynomial equation.

For example, 2x² + 3x + 1 is a polynomial. When we set it to zero, it becomes a polynomial equation: 
2x² + 3x + 1 = 0

Look at more such equations:

• x² – 4 = 0
• 3x³ + x – 7 = 0
• x⁴ + 2x² – x + 5 = 0
   

A polynomial is an algebraic expression made up of variables and constants. We use addition, subtraction, and multiplication to work with polynomials. The variables in a polynomial have powers that are non-negative integers.
Example: 2x + 3

A polynomial equation is formed when a polynomial is made equal to a value using the equal sign (=).
Example: 2x + 3 = 0

Note:

• The degree of a polynomial equation depends on the highest exponent of the variable. 
• To solve the equation, we find the value or values of the variable that make both sides equal.
• If you know the solutions, you can also form a polynomial equation from them. 
   

A polynomial equation is in the form:
p(x) = 0

In algebra, it can be written in the form:
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0

Where:

• aₙ, aₙ₋₁, ..., a₁, a₀ are real numbers called coefficients
• ‘x’ is the variable
• p(x) means "a polynomial in x"
• n is a non-negative integer that shows the degree of the polynomial, which is the highest exponent of x in the expression.
   

Depending on the degree of a polynomial equation, it is classified into different types. We will learn about each one step by step:

Linear Polynomial Equation

Linear polynomials are equations where the variable has the highest power of 1.
General Form: ax + b = 0

Examples:
  4x + 5 = 0
  7x – 9 = 0

Quadratic Polynomial Equation

Quadratic polynomial equations are equations where the variable has the highest power of 2.

General Form: ax² + bx + c = 0

Examples:
  2x² – 4x + 6 = 0
  5x² + 3x – 8 = 0

Cubic Polynomial Equation

If the highest power of the variable is 3, then it is said to be a cubic polynomial equation.

General Form: ax³ + bx² + cx + d = 0

Examples:
  x³ + 2x² – 3 = 0
  2x³ – x + 4 = 0

Biquadratic Polynomial Equation

Biquadratic polynomial equations are polynomial equations in which the highest power of the variable is 4.

General Form: ax⁴ + bx³ + cx² + dx + e = 0

Example: 4x⁴ + 2x² – 6x + 1 = 0.
 

To solve the equation p(x) = 0, we have to find the value or values of x that satisfy the equation. Such values are called roots or zeros because p(a) = 0, where ‘a’ is a root of the polynomial. Let’s learn how different polynomial equations are solved.

Forming Linear Polynomial Equations

To solve a linear polynomial equation, we set it equal to zero and find the value of the variable.
Example:
Solve 6x - 9 = 0

Step 1: Add 9 to both sides:
 6x = 9

Step 2: Divide both sides by 6:
 x = 9/6 = 3/2

Solving a Quadratic Polynomial from its Roots

To find a quadratic polynomial when its roots are known, we utilize the fact that the polynomial can be formed using the sum and product of the roots.
Find the quadratic polynomial whose roots are -3 and 5.

Step 1: Sum of roots = -3 + 5 = 2

Step 2: Product of roots = -3 × 5 = -15

Step 3: Use the formula
 x² - (sum of roots)x + (product of roots) = 0
 Substitute values:
 x² - 2x - 15 = 0

Solving Cubic Polynomial Equations

To solve a cubic polynomial equation, we find its roots by factoring or by using techniques such as synthetic division and the factor theorem. Let’s solve it step by step.

General form:
ax³ + bx² + cx + d = 0, a ≠ 0

Steps:

1.     Write the equation in standard form.
2.      Use the factor theorem or synthetic division to factor into a linear factor and a quadratic factor.
3.      Find the remaining roots by solving the quadratic equation

Example:

Solve x³ - 5x² + 8x - 4 = 0.

Step 1: The given equation is already in standard form.

Step 2: Substitute x = 1 as a root:
1 – 5 + 8 – 4 = 0, so x = 1 is a root.
 

Polynomial equations have several practical applications in different fields and help students develop problem-solving skills. Let’s now learn how they are applied in various real-life situations:

• In business and economics, polynomial equations are used to analyze and predict profit, revenue, and costs for forecasting and decision-making.

• In animation and video games, polynomials help to design smooth curves and realistic motion paths.

• Polynomial equations in environmental studies are used to track changes in pollution levels, population growth, and climate patterns over a period.