Polynomials in One Variable

Polynomials in one variable are algebraic expressions consisting of terms with single variable and coefficients, combined using arithmetic operations. For example:
x2 + 5x + 8
2y3 + 5y2 + 9y + 3  
The above examples have only one variable (‘x’ in the first example and ‘y’ in the second).
 

Polynomials in one variable can be grouped into different types, depending on their degree. The degree of a polynomial is simply the highest exponent of the variable. Based on this, polynomials are usually divided into four main types: 

• Zero or Constant Polynomial
• Linear Polynomial
• Quadratic Polynomial
• Cubic Polynomial 

Zero or Constant Polynomial: A zero or constant polynomial’s variable has no exponent, meaning its degree is 0. As a result, these polynomials only include constant terms. For example, 2x0 = 21 = 2.

Linear Polynomial: The polynomials with 1 as the highest degree of the variable are the linear polynomials. For example, x + 2, 6x + 5. 

Quadratic Polynomial: Quadratic polynomials are the polynomials with the highest degree of 2. For example, 5x2 + 2x + 6, 6y2 + 5y + 1

Cubic Polynomial: A cubic polynomial is one in which the highest degree of the polynomial is 3. For example, 5x3 + 2x2 + 8x + 7
 

Solving a polynomial is the process of finding the values of the variable that make the whole expression equal to zero. The solution of the polynomial is also known as roots or zeros of the polynomial. A polynomial of degree n can have up to n roots. So, linear polynomials have one root, quadratic polynomials have two roots and cubic polynomials have three roots. 

Solving Linear Polynomial in One Variable

The general form of a linear polynomial is ax + b = 0. Now let’s learn how to solve a linear polynomial, with an example:
3x - 6 = 0
Step 1: To solve, isolate the term with the variable
3x = 6
Step 2: Isolate the variable
x = 6/3
x = 2
 

To solve a quadratic polynomial in one variable, there are different methods such as: 

• Factoring
• Using quadratic formula
• Completing the square

Factoring: In the factoring method, the quadratic polynomial is expressed as a product of two binomial expressions. The general form of a quadratic equation is ax2 + bx + c = 0. To factor, we find two numbers whose product is equal to ac and sum is equal to b. 
For example, finding the root of x2 + 5x + 6 = 0
Here, a = 1, b = 5, and c = 6
Here the factors are 2 and 3, whose sum is 5 and product is 6
So, x2 + 5x + 6 = (x + 2)(x + 3) = 0
x + 2 = 0 and x + 3 = 0
x = -2 and x = -3

Using Quadratic Formula: The general form of a quadratic polynomial is ax2 + bx + c = 0, where a ≠ 0. The formula to find the value of x is x = -b ± b2 - 4ac2a. 
For example, 2x2 - 4x - 6 = 0
Here, a = 2, b = -4 and c = -6
x = -(-4) ± (-4)2 - 4 × 2 × (-6)2 × 2  
x = 4 ± 16 + 484
x = 4 ± 644
x = 4 ± 84
x = 4 + 84 and x = 4 - 84
x = 124 and x = -44
x = 3 and x = -1

Solving Cubic Polynomial in One Variable

The general form of a cubic equation is ax3 + bx2 + cx + d = 0, where a ≠ 0. To solve a cubic equation, follow these steps: 
For example, finding the value of x in x3 - 6x2 + 11x - 6 = 0
Here, the equation is in standard form: x3 - 6x2 + 11x - 6 = 0
If x = 1: x3 - 6x2 + 11x - 6 = 0
1 – 6(1)2 + 11(1) - 6 = 0
1 – 6 + 11 – 6 = 0
So, x = 1 is a root
Factoring: (x -1)(x2 - 5x + 6)
Solving the quadratic equation: x2 - 5x + 6
x2 - 5x + 6 = (x - 2)(x - 3) = 0
So, x = 1, 2, 3
 

Polynomials in one variable are useful in various fields, such as business, science, and engineering, for modeling situations, making predictions, and analyzing patterns. In this section, we will learn some real-world applications of polynomials. 

• In physics, polynomials in one variable are used in modeling projectile motion. It helps to calculate the trajectory of a ball, a rocket, or a jet of water from a fountain. 
•  In civil engineering, a polynomial in one variable is used to design roller coasters to ensure smooth and continuous curves to ensure a safe and exciting ride for riders. 
• In finance, to calculate the compound interest over several periods, financial planning, retirement saving projections, and loan calculations we use the polynomials in one variable.