Polynomials Lesson: Definitions, Types, Operations & Example

Lesson Overview

• What Is a Polynomial?
• Types of Polynomials
• Operations on Polynomials
• How to Solve Polynomials
• 1. Factoring Polynomials
• 2. Using the Quadratic Formula
• 3. Using Synthetic Division
• Examples of Polynomials

Polynomials are a fundamental concept in algebra, representing expressions made up of terms that follow specific rules. From simple linear expressions to complex quadratic forms, polynomials are essential in solving equations and modeling real-world scenarios. For example, expressions like "11x + 9" or "x² + 11x + 28" are classified as polynomials.

What Is a Polynomial?

A polynomial is a mathematical expression comprising variables, coefficients, and exponents, combined using addition, subtraction, and multiplication.

Fig: Structure of a Polynomial

A polynomial can have one or more terms. These terms are separated by addition or subtraction. Here are a couple of examples:

• 3x² + 2x + 1: This is a polynomial with three terms.
• 5y³ - y: This is another polynomial but with two terms.

The degree of polynomial is defined as the highest exponent of the variable in the expression. For example, in the polynomial:

4x³ + 2x² - x + 7, the highest exponent is 3 (from the term 4x³), so the degree of this polynomial is 3.

Polynomials must not contain division by variables (1/x​) or negative exponents (such as x⁻²), as these would violate the definition of a polynomial. Polynomials can only have non-negative integer exponents.

Types of Polynomials

Polynomials are classified based on two main characteristics: the number of terms and the degree (the highest power of the variable). Below are the main types of polynomial equations, categorized by these factors.

Type of Polynomial	Degree	Description	Example
Constant Polynomial	0	A polynomial with no variables, just a constant number.	5
Linear Polynomial	1	A polynomial where the highest power of the variable is 1. Graphs as a straight line.	3x + 2
Quadratic Polynomial	2	A polynomial where the highest power of the variable is 2. Graphs as a parabola.	x² - 4x + 4
Cubic Polynomial	3	A polynomial where the highest power of the variable is 3. Can have more complex shapes when graphed.	2x³ - 3x² + x - 1

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Operations on Polynomials

Polynomials can undergo several operations, following specific rules for combining or manipulating the terms of polynomials.

1. Addition of Polynomials

To add polynomials, combine like terms. Like terms are terms that have the same variable raised to the same power. You add the coefficients of these like terms.

Example:

(3x² + 2x) + (4x² - x) = 7x² + x

In this example:

• Combine the terms with x²: 3x² + 4x² = 7x²
• Combine the terms with x: 2x - x = x

So, the sum is 7x² + x.

2. Subtraction of Polynomials

To subtract polynomials, subtract the corresponding terms. Be careful to distribute the negative sign across the second polynomial.

Example:

(5x² + x) - (3x² - 2x) = 2x² + 3x
In this example:

Subtract the terms with x: x - (-2x) = x + 2x = 3x

Subtract the terms with x²: 5x² - 3x² = 2x²

So, the result is 2x² + 3x

3. Multiplication of Polynomials

When multiplying polynomials, use the distributive property. Multiply each term in the first polynomial by every term in the second polynomial, then combine the results.

Example:

(x + 2)(x - 3) = x(x - 3) + 2(x - 3)
Now, distribute each term:

• x(x - 3) = x² - 3x
• 2(x - 3) = 2x - 6

Now, combine all terms:
x² - 3x + 2x - 6 = x² - x - 6

So, the product is x² - x - 6.

4. Division of Polynomials

For polynomial division, you can use polynomial long division or synthetic division (in cases involving linear divisors). This process involves dividing the terms of the numerator by the terms of the denominator.

Example:

Dividing x² + 3x + 2 by x + 1:

1. Divide the first term (x²) by the first term of the denominator (x), which gives x.
2. Multiply x by (x + 1), resulting in x² + x.
3. Subtract (x² + x) from (x² + 3x + 2), leaving 2x + 2.
4. Divide 2x by x, which gives 2.
5. Multiply 2 by (x + 1), resulting in 2x + 2.
6. Subtract (2x + 2) from (2x + 2), leaving a remainder of 0.

So, the division of x² + 3x + 2 by x + 1 equals x + 2.

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How to Solve Polynomials

Solving polynomials means finding the values of the variable that make the polynomial equation equal to zero. Here are three common polynomial operations methods for solving polynomials:

1. Factoring Polynomials

Factoring is often the first method you should try when solving polynomials. The goal is to express the polynomial as a product of simpler polynomials and then use the Zero Product Property. This property states that if a product of factors equals zero, at least one of the factors must be zero.

Steps to Factor a Polynomial:

• Look for a common factor in all terms and factor it out.
• If the polynomial is quadratic (degree 2), try factoring it into two binomials.
• If the polynomial is cubic or higher, look for special patterns, such as the difference of squares, perfect square trinomials, or use grouping.

Example: Solve x² - 5x + 6 = 0.

1. Factor the polynomial: (x - 2)(x - 3) = 0
2. Set each factor equal to zero: (x - 2) = 0 or (x - 3) = 0
3. Solve for x: x = 2 or x = 3

So, the solutions are x = 2 and x = 3.

2. Using the Quadratic Formula

For quadratic polynomials (degree 2) that are difficult to factor, you can use the quadratic formula to find the solutions. The quadratic formula works for any quadratic equation of the form ax² + bx + c = 0.

The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a

Where:

• a, b, and c are the coefficients of the quadratic equation.
• The discriminant (b² - 4ac) determines the nature of the roots:
  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there is one real double root.
  • If the discriminant is negative, there are two complex roots.

Example: Solve x² + 4x + 4 = 0.

1. Identify the coefficients: a = 1, b = 4, and c = 4.
2. Substitute into the quadratic formula: x = (-4 ± √(4² - 4(1)(4))) / (2(1)) = (-4 ± √(16 - 16)) / 2 = (-4 ± √0) / 2
3. Simplify the expression: x = -4 / 2 = -2

So, the solution is x = -2.

3. Using Synthetic Division

For higher-degree polynomials, synthetic division is a helpful method for dividing the polynomial by a linear factor, such as x - a. This method is often used to find factors of the polynomial, which can then be solved.

Steps for Synthetic Division:

• Set up the synthetic division table with the coefficients of the polynomial.
  
• Divide by the root of the factor (for example, if dividing by x - 2, use 2).
  
• If the remainder is zero, the factor x - a is valid, and you can continue factoring the quotient.

Example: Solve x³ - 4x² + 3x - 2 = 0 using synthetic division, assuming x - 1 is a factor.

1. Set up the synthetic division:

2. The quotient is x² - 3x with a remainder of zero, confirming that x - 1 is a factor.

3. Factor the quadratic: x² - 3x = x(x - 3)

4. So, the factored form is (x - 1)(x)(x - 3) = 0.

5. Set each factor equal to zero: x - 1 = 0 → x = 1 x = 0 x - 3 = 0 → x = 3

The solutions are x = 1, x = 0, and x = 3.

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Examples of Polynomials

Here are a few polynomial examples, with solutions for each type, to help you understand the different forms and how they can be solved or simplified.

Example 1: Linear Polynomial

A linear polynomial is a polynomial of degree 1, with the highest exponent of the variable being 1.

Example:
Solve the equation 3x + 4 = 0.

Steps to solve:

1. Start with the equation:
   3x + 4 = 0
2. Subtract 4 from both sides:
   3x = -4
3. Divide both sides by 3 to isolate x:
   x = -4 / 3

Solution:
x = -4/3

Example 2 : Quadratic Polynomial

A quadratic polynomial is a polynomial of degree 2, where the highest power of the variable is 2.

Example:
Solve the equation x² - 5x + 6 = 0.

Steps to solve:

1. Factor the quadratic polynomial:
   (x - 2)(x - 3) = 0

2. Set each factor equal to zero:
   x - 2 = 0 → x = 2
   x - 3 = 0 → x = 3

Solution:
x = 2 and x = 3

Example 3: Cubic Polynomial

A cubic polynomial is a polynomial of degree 3, where the highest power of the variable is 3. These polynomials often have more than one solution.

Example:
Solve the equation x³ - 3x² - 4x = 0.

Steps to solve:

1. Factor the polynomial:
   x(x² - 3x - 4) = 0

1. Now factor the quadratic part:
   x(x - 4)(x + 1) = 0

Set each factor equal to zero:
x = 0
x - 4 = 0 → x = 4
x + 1 = 0 → x = -1

Solution:
x = 0, x = 4, and x = -1

Example 4: Quartic Polynomial

A quartic polynomial is a polynomial of degree 4. These polynomials can have up to four solutions, depending on the nature of the factors.

Example:
Solve the equation x⁴ - 5x² + 4 = 0.

Steps to solve:

1. Let y = x², so the equation becomes:
   y² - 5y + 4 = 0

2. Factor the quadratic equation in y:
   (y - 1)(y - 4) = 0

3. Now, substitute back y = x²:
   (x² - 1)(x² - 4) = 0

4. Factor both terms:
   (x - 1)(x + 1)(x - 2)(x + 2) = 0

Set each factor equal to zero:
x - 1 = 0 → x = 1
x + 1 = 0 → x = -1
x - 2 = 0 → x = 2
x + 2 = 0 → x = -2

Solution:
x = 1, x = -1, x = 2, and x = -2

Example 5: Polynomial with More Than One Variable

Polynomials can also have more than one variable. These are often used in higher dimensions or to represent more complex relationships.

Example:
Solve the equation 2x² + 3xy - y² = 0, where x and y are variables.

Steps to solve:

1. Factor the polynomial by grouping:
   (2x² + 3xy) - (y²) = 0

2. Factor each group:
   x(2x + 3y) - y(2x + 3y) = 0

3. Factor out the common binomial factor:
   (x - y)(2x + 3y) = 0

4. Set each factor equal to zero:
   x - y = 0 → x = y
   2x + 3y = 0 → 2x = -3y → x = -3y/2

Solution:
The solutions are x = y and x = -3y/2.