Polynomials| Degree | Types | Properties and Examples

Polynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.

A polynomial can have one or several terms. Each term is a product of a constant and a variable raised to an exponent.

For Example:

• x²,
• x² + 2x,
• 5x + y,
• 3x³ + 5x² - 4x,
• √2(x) + y, etc.

In the image added below, we have shown a polynomial, with variables, constants, and a leading coefficient:

Degree of Polynomial

Degree of polynomial is determined by the highest exponent or power of the variable in the expression. It represents the highest degree term in the polynomial.

To find the degree of polynomial there are two cases:

Degree of Single Variable Polynomial

The degree of a single variable polynomial is determined by the highest exponent or power of that variable in the polynomial expression. It is represented by the highest degree term in the polynomial.

Example: 5x⁵ - 2x³ + 4x² - x + 4
Solution:

The highest exponent is 5, which corresponds to the term 5x⁵. Therefore, the degree of this polynomial is 5.

Note: The degree of any polynomial which contains constant terms will be zero.

Degree of a Multivariable Polynomial

The degree of a multivariable polynomial is determined by the sum of the exponents of the variables in each term, considering the highest sum among all the terms.

Example: 5x²y³ - 2xy⁴ + 3x³+ 10
Solution:

In this expression, we have terms with variables x and y raised to different powers. To find the degree, we calculate the sum of the exponents for each term:

• First term, 5x²y³, has a sum of exponents equal to 2 + 3 = 5.
• Second term, -2xy⁴, has a sum of exponents equal to 1 + 4 = 5.
• Third term, 3x³, has a sum of exponents equal to 3.

Among all the terms the highest sum of exponents is 5. Therefore, the degree of this polynomial is 5.

Types of Polynomials

Polynomials can be categorized into several types based on the number of terms they contain or their degree.

Based on the Number of Terms:

1. Monomial:
   • A polynomial with just one term. For example, 7x³ or −4.
2. Binomial:
   • A polynomial with two terms. For example, x² − 5x².
3. Trinomial:
   • A polynomial with three terms. For example, 2x² − 3x + 4.
4. Multinomial:
   • A polynomial with more than three terms. For example, x⁴ − 2x³ + 3x² − x + 1.

Based on Degree:

1. Constant Polynomial:
   • A polynomial of degree 0. It has no variable terms, just a constant. For example, 5 or −3.
2. Linear Polynomial:
   • A polynomial of degree 1. It forms a straight line when graphed. For example, 3x + 2.
3. Quadratic Polynomial:
   • A polynomial of degree 2. It forms a parabolic shape when graphed. For example, x² − 4x + 4.
4. Cubic Polynomial:
   • A polynomial of degree 3. It can have points of inflection and typically has the shape of an S-curve when graphed. For example, x³ − 6x² + 11x − 6.

Read More: Types of Polynomials (Based on Terms and Degrees)

Standard Form of a Polynomial

P(x) = a_nxⁿ + a_n−1 xⁿ⁻¹ + ⋯ + a₁ x + a₀​

• a_n, a_n−1, …, a₁, a₀ are coefficients (which can be real or complex numbers).
• x is the variable.
• n is a non-negative integer representing the degree of polynomial.
• a_n is the leading coefficient and a₀ is the constant term.

Each term consists of a variable raised to a non-negative whole number power, multiplied by a number.

Example: Express the variable in Standard form: x² - 3x³ + 4x

Given Polynomial: x² - 3x³ + 4x

In standard form polynomial is arranged in decreasing order of their degree = -3x³ + x² + 4x

Terms in a Polynomial

In a polynomial various terms are separated using the plus (+) and the minus (-) sign. Depending on the number of terms in a polynomial a polynomial can be monomial, binomial, etc.

In a polynomial, we classify various terms into like terms and unlike terms.

• Like terms are the terms that have the same variable and with same exponent.
• Unlike terms are terms that have different variables or different powers(even with the same variable).

Read more about Like and Unlike Terms.

For example, Identify like terms and unlike terms in the polynomial, 4x³ + 2x² -3xy + x³ - xy

Given Polynomial : 4x³ + 2x² -3xy + x³ - xy

Like Terms: (4x³, x³) and (-3xy, -xy)
Unlike Terms: (4x³, x²), (x³, x²), (-3xy, x²), (-xy, x²), etc

Properties of Polynomials (Theorems of Polynomials)

Various properties of the polynomials are:

• For two polynomials P(x) and Q(x) :
  Degree (P ± Q) ≤ max(deg⁡ P, deg ⁡Q)
  Deg⁡(P⋅Q) = deg⁡ P + deg⁡ Q

• For two given polynomials P(x) and Q(x) we always have unique polynomials Q (Quotient Polynomial) and R (Residue Polynomial) such that, P = R.Q + R
• If a polynomial P(x) is divisible by polynomial x - a, then P(a) = 0 is always true. This is also called Bezout's Theorem or Factor Theorem.
• If a polynomial Q divides the polynomial P then zeros of polynomial Q are the zeros of polynomial P.
• For a polynomial of n degree, we have n roots that may be real or complex. The remainder of a polynomial f(x) when divided by (x - a) is f(a).

Solving Polynomials

Solving polynomials or solving polynomial equations means finding the roots or zeros of polynomials which are the values of the variable that make the equation true for a specific polynomial equation. There are various methods to find the root of the polynomial equations which often involve factorization of polynomials.

Zeroes of Polynomials: Zeros of a polynomial, also known as roots or solutions, are the values of the variable that make the polynomial equal to zero. In other words, they are the values of x for which the polynomial evaluates to zero.

If x is a zero of a given polynomial P(x) then p(x) = 0

Solving polynomial equations is a foundational skill in algebra and it is used in fields ranging from engineering to economics, where relationships defined by polynomials need to be analyzed and used.

Example: Solve x² − 5x + 6 = 0
Solution:

Step 1: Set the Equation to Zero : Ensure your polynomial equation is in the standard form, where one side is zero. x² − 5x + 6 = 0

Step 2: Factor the Polynomial : Factorize the polynomial into simpler binomial factors, if possible.

x² − 5x + 6 = (x−2)(x−3).

Step 3: Apply the Zero Product Property : According to the zero product property, if a product equals zero, then at least one of the factors must be zero.

(x - 2) = 0 and (x - 3) = 0

Step 4: Solve for the Variable : Solve each equation from the previous step:

• From (x - 2) = 0, x = 2.
• From (x − 3) = 0, x = 3.

Step 5: Check Your Solutions : Substitute your solutions back into the original equation to verify:

• Put x=2 in x² − 5x + 6 = 0 ⇒ (2)² − 5(2) + 6 = 4 − 10 + 6 = 0
• Put x = 3 in x² − 5x + 6 = 0⇒(3)² − 5(3) + 6 = 9 − 15 + 6 = 0.

Articles related to Polynomials:

• Algebraic Expressions
• Factoring Polynomials
• Polynomial Formula
• Adding and Subtracting Polynomials

Solved Examples of Polynomials

Question 1: Find the value of x in the polynomial equation 2x² - 5x + 3 = 0.
Solution:

We can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a

where a = 2, b = -5, and c = 3.

Substituting the values:

x = (5 ± √(25 - 423)) / 2*2
x = (5 ± √(25 - 24)) / 4
x = (5 ± √1) / 4

So, the solutions are:

x = (5 + 1) / 4 = 6 / 4 = 3/2 and x = (5 - 1) / 4 = 4 / 4 = 1.

Question 2: Factorize the polynomial x² - 4x + 4.
Solution:

We observe that the given polynomial is a perfect square trinomial.
It can be written as (x - 2)².

So, the factored form is (x - 2)(x - 2) or (x - 2)².

Question 3: Given the polynomial 3x⁴ - 7x³ + 2x² - 5x + 1, find its degree and leading coefficient.
Solution:

The degree of a polynomial is the highest power of the variable present. In this case, the degree is 4. The leading coefficient is the coefficient of the term with the highest power of the variable. Here, the leading coefficient is 3.

Question 4: Simplify the expression (2x² - 3x + 1)(x² + 4x - 2).
Solution:

We use the distributive property to expand the expression:

(2x² - 3x + 1)(x² + 4x - 2) = 2x²(x² + 4x − 2) − 3x(x² + 4x − 2) + 1(x² + 4x − 2)
= 2x⁴ + 8x³ − 4x² − 3x³ − 12x² + 6x + x² + 4x − 2
= 2x⁴ + 5x³ - 15x² + 10x - 2

Also Read:

• Polynomials in One Variable
• Polynomial formula
• Factorization of Polynomials