Polynomials: Types, Equations, Degrees & Theorems

Polynomials are expressions that contain constants, variables, exponents, and terms. Polynomials are important mathematical expressions in algebra. They have several applications in mathematics as well as physics. Several operations such as addition, subtraction, and multiplication can be performed on polynomials. Based on that, the division algorithm, factor theorem, and remainder theorem are proved.

---

Polynomials

[Click Here for Sample Questions]

Polynomials are defined as expressions that contain constants, variables, exponents, and terms. The term polynomials is derived from the words 'poly' meaning 'many' and 'nomial' meaning terms. Thus, a polynomial is something that has many terms.

The general expression of polynomials is

a₀xⁿ + a₁x^n-1 + a₂x_n-2 + … an

Where,

n is a non-negative integer

x is a variable

a₀, a₁, a₂, …, an are real numbers

Some examples of Polynomials are:

x² + 6x + 9;

2x + 5;

x² + 4y + 2;

x³ + y³ + 3xy(x + y);

w + x + y + z;

The video below explains this:

Polynomials Detailed Video Explanation:

Read More:

---

Types Of Polynomials

[Click Here for Sample Questions]

Every expression in a polynomial is known as a term of that polynomial. For example, in the polynomial x² + 4y + 2, the expressions x², 4y, and 2 are each term of that polynomial.

A polynomial should have at least one term. No polynomials can have zero terms. The terms of a polynomial can be constants, variables, or a mixture of constants and variables.

• Monomials: Polynomials containing a single term are called monomials. Example: 2x
• Binomials: Polynomials containing two terms are called binomials. Example: 2x + y
• Trinomials: Polynomials containing three terms are called trinomials. Example: 2x² + 4y + 2

Read More:

---

Important Terms Related to Polynomial

[Click Here for Sample Questions]

Some of the important terms related to polynomials are as follows:

• Coefficients Of Polynomials: Every term in polynomials has a coefficient. In the polynomial x2 + 4y + 2, the terms are x², 4y, and 2. The coefficient of x² is 1 and of y is 4. Thus, for a polynomial ax² + bx + c, a and b are coefficients where at least one of the coefficients is non-zero. 
• Constants Of Polynomials: In polynomials, the real numbers without any variables are called constants. For example, in the polynomial x² + 4y + 2, 2 is the constant. A polynomial having no variables is called a constant polynomial. 32 is a constant polynomial.
• Exponents Of Polynomials: Exponents of polynomials are defined as the power to which the variables are raised. Example: In polynomial x², 2 is the exponent.

Also Read:

---

Degree Of A Polynomial

[Click Here for Sample Questions]

The degree of a polynomial is defined as the highest power of the polynomial. For example, in the polynomial x² + 4y + 2, 2 is the degree of that polynomial. And in the polynomial x3 + y3 + 3xy(x + y), 3 is the degree of the polynomial. For a constant polynomial, the degree of the polynomial is zero.

Also Read: Permutations and Combinations 

Based on the degree of polynomials, the polynomials can be classified as linear, quadratic, cubic, etc.

• Linear Polynomials: The polynomials having the degree of polynomial one are called linear polynomials. Example: 2x + 5
• Quadratic Polynomials: The polynomials having the degree of polynomial two are called quadratic polynomials. Example: x2 + 6x + 9
• Cubic Polynomials: The polynomials having the degree of polynomial three are called cubic polynomials. Example: x3 + y3 + 3xy(x + y)
• Polynomials In One Variable: Polynomials in one variable are defined as the polynomials that have only one type of variable in all of its terms. Example: 3x + 2, 5x2 + 3x + 7, etc.

Also Read:

---

Zeros Of Polynomials

[Click Here for Sample Questions]

In a polynomial p(x), a real number 'a' is called zero of that polynomial if p(a) = 0.

Zeros of polynomials follow the rules of algebraic identities.

Sum Of Zeros Of Polynomials

For a quadratic polynomial ax² + bx + c = 0, if α and β are the two zeros of that polynomial then the sum of the zeros of the polynomial is given as,

α + β = -ba = −coefficient of x . coefficient of x²

For a cubic polynomial ax³ + bx² + cx + d = 0, if α, β, and γ are the three zeros of that polynomial then the sum of the zeros of the polynomial is given as,

α + β + γ = -ba = −coefficient of x² . coefficient of x³

Also,

αβ + βγ + γα = ca = coefficient of x . coefficient of x³

Also Read: Types of Probability

Product Of Zeros Of Polynomials

For a quadratic polynomial ax² + bx + c = 0, if α and β are the two zeros of that polynomial then the product of the zeros of the polynomial is given as,

αβ = ca = constant term . coefficient of x2

For a cubic polynomial ax³ + bx² + cx + d = 0, if α, β, and γ are the three zeros of that polynomial then the product of the zeros of the polynomial is given as,

αβγ = -da = -constant term coefficient of x³

---

Division Algorithm

[Click Here for Sample Questions]

It is known that,

Dividend = Divisor x Quotient + Remainder

Therefore, for two polynomials p(x) and g(x) where g(x) ≠ 0, the relationship between these two polynomials and their remainder r(x) and quotient q(x) is given as,

p(x) = g(x) × q(x) + r(x)

Also Read:

---

Remainder Theorem

[Click Here for Sample Questions]

Remainder Theorem states that if any polynomial p(x) has the degree of polynomial greater than or equal to one and if this polynomial is divided by a linear polynomial (x - a) then the remainder of this division is p(a).

Proof: Let p(x) be any polynomial with the degree of polynomial greater than or equal to 1.

Suppose, p(x) is divided by (x – a), and the quotient is q(x) and the remainder is r(x).

Therefore, by Division Algorithm,

p(x) = (x – a) x q(x) + r(x) …(1)

Since the degree of (x – a) is 1 and the degree of r(x) is less than the degree of (x – a), the degree of r(x) = 0.

This means that r(x) is a constant. Let us assume this constant to be r.

Thus, for every value of x, r(x) = r.

Therefore, we can write equation (1) as,

p(x) = (x – a) x q(x) + r

If one assumes that x = a, then the above equation can be written as

p(a) = (a – a) x q(a) + r = r

This proves the theorem.

Also Read: 

---

Factorisation Of Polynomials

[Click Here for Sample Questions]

From the Division Algorithm, it is known that p(x) = g(x) × q(x) + r(x). 

But if the remainder is zero i.e. r(x) = 0 then

p(x) = g(x) × q(x)

This proves that a single polynomial can be expressed as a product of two or more polynomials.

The process of obtaining two or more polynomials from a single polynomial such that their products give the original polynomial is known factorisation of polynomials.

Example: 6p² + 3p = 3p x (2p + 1)

Here 3p and (2p + 1) are factors of 6p² + 3p.

Also Read: Complex numbers and Quadratic equations

---

Factor Theorem

[Click Here for Sample Questions]

Factor Theorem states that “for a polynomial p(x), (x - c) is its factor if and only if p(c) = 0”.

Also, if (x - c) is a factor of a polynomial p(x) then p(c) = 0.

Proof: It is known that Remainder Theorem gives us the equation

p(x) = (x – a) x q(x) + p(a)

If one assumes that p(a) = 0, then the above equation can be written as,

p(x) = (x – a) x q(x)

This proves that (x - a) is a factor of p(x).

Also,

If (x - a) is a factor of p(x), then

p(x) = (x – a) x g(x)

Where,

 g(x) is a polynomial which is a factor of p(x).

If one assumes that x = a then the above equation can be written as,

p(x) = (a – a) x g(x)

p(x) = 0

This proves the Factor Theorem.

Also Read:

---

Methods Of Factorisation Of Polynomials

[Click Here for Sample Questions]

Factorisation of polynomials can be done with a number of methods. Some of them are:

1. Factorisation by dividing the polynomial by its Highest Common Factor (HCF).
2. Factorisation by grouping the terms in the polynomial.
3. Factorisation by use of identities.

Important Identities:

• (x + y)² = x² + 2xy + y²
• (x – y)² = x² – 2xy + y²
• x² – y² = (x + y) (x – y)
• (x + a) (x + b) = x2 + (a + b)x + ab
• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
• (x + y)³ = x³ + y³ + 3xy(x + y)
• (x – y)³ = x³ – y³ – 3xy(x – y)
• x³ + y³ + z³ – 3xyz = (x + y + z) (x² + y² + z² – xy – yz – zx)

Also Read: 

---

Things to Remember

[Click Here for Sample Questions]

• Polynomials are defined as expressions that contain constants, variables, exponents, and terms.

The general expression of polynomials is

a₀xn + a¹xn^-1 + a²x^n-2 + … aⁿ

• Every polynomial has at least one constant, coefficient, and exponent.
• The degree of a polynomial is the highest power to which the variable term is raised in the polynomial.
• Dividend = Divisor x Quotient + Remainder
• The Remainder Theorem states that if any polynomial p(x) has the degree of polynomial greater than or equal to one and if this polynomial is divided by a linear polynomial (x - a) then the remainder of this division is p(a).
• Factor Theorem states that for a polynomial p(x), (x - c) is its factor if and only if p(c) = 0. Also, if (x - c) is a factor of a polynomial p(x) then p(c) = 0.

Also Read: