Zeros of a polynomial.

Degree of a Polynomial

The degree of a polynomial is defined as the highest degree of a monomial within a polynomial. Thus, a polynomial equation having one variable which has the largest exponent is called a degree of the polynomial.

Polynomial	Degree	Example
Constant or Zero Polynomial	0	6
Linear Polynomial	1	3x+1
Quadratic Polynomial	2	4x2+1x+1
Cubic Polynomial	3	6x3+4x3+3x+1
Quartic Polynomial	4	6x4+3x3+3x2+2x+1

Example: Find the degree of the polynomial 6s⁴+ 3x²+ 5x +19

Solution:

The degree of the polynomial is 4.

Terms of a Polynomial

The terms of polynomials are the parts of the equation which are generally separated by “+” or “-” signs. So, each part of a polynomial in an equation is a term. For example, in a polynomial, say, 2x² + 5 +4, the number of terms will be 3. The classification of a polynomial is done based on the number of terms in it.

Polynomial	Terms	Degree
P(x) = x3-2x2+3x+4	x3, -2x2, 3x and 4	3

Types of Polynomials

Polynomials are of 3 different types and are classified based on the number of terms in it. The three types of polynomials are:

• Monomial
• Binomial
• Trinomial

These polynomials can be combined using addition, subtraction, multiplication, and division but is never division by a variable. A few examples of Non Polynomials are: 1/x+2, x^-3

Monomial

A monomial is an expression that contains only one term. For an expression to be a monomial, the single term should be a non-zero term. A few examples of monomials are:

• 5x
• 3
• 6a⁴
• -3xy

Binomial

A binomial is a polynomial expression that contains exactly two terms. A binomial can be considered as a sum or difference between two or more monomials. A few examples of binomials are:

• – 5x+3,
• 6a⁴ + 17x
• xy²+xy

Trinomial

A trinomial is an expression that is composed of exactly three terms. A few examples of trinomial expressions are:

• – 8a⁴+2x+7
• 4x² + 9x + 7

Monomial	Binomial	Trinomial
One Term	Two terms	Three terms
Example: x, 3y, 29, x/2	Example: x2+x, x3-2x, y+2	Example: x2+2x+20

Properties

Some of the important properties of polynomials along with some important polynomial theorems are as follows:

Property 1: Division Algorithm

If a polynomial P(x) is divided by a polynomial G(x) results in quotient Q(x) with remainder R(x), then,

P(x) = G(x) • Q(x) + R(x)

Property 2: Bezout’s Theorem

Polynomial P(x) is divisible by binomial (x – a) if and only if P(a) = 0.

Property 3: Remainder Theorem

If P(x) is divided by (x – a) with remainder r, then P(a) = r.

Property 4: Factor Theorem

A polynomial P(x) divided by Q(x) results in R(x) with zero remainders if and only if Q(x) is a factor of P(x).

Property 5: Intermediate Value Theorem

If P(x) is a polynomial, and P(x) ≠ P(y) for (x < y), then P(x) takes every value from P(x) to P(y) in the closed interval [x, y].

Property 6

The addition, subtraction and multiplication of polynomials P and Q result in a polynomial where,

Degree(P ± Q) ≤ Degree(P or Q)

Degree(P × Q) = Degree(P) + Degree(Q)

Property 7

If a polynomial P is divisible by a polynomial Q, then every zero of Q is also a zero of P.

Property 8

If a polynomial P is divisible by two coprime polynomials Q and R, then it is divisible by (Q • R).

Property 9

If P(x) = a₀ + a₁x + a₂x² + …… + a_nxⁿ is a polynomial such that deg(P) = n ≥ 0 then, P has at most “n” distinct roots.

Property 10: Descartes’ Rule of Sign

The number of positive real zeroes in a polynomial function P(x) is the same or less than by an even number as the number of changes in the sign of the coefficients. So, if there are “K” sign changes, the number of roots will be “k” or “(k – a)”, where “a” is some even number.

Property 11: Fundamental Theorem of Algebra

Every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Property 12

If P(x) is a polynomial with real coefficients and has one complex zero (x = a – bi), then x = a + bi will also be a zero of P(x). Also, x² – 2ax + a² + b² will be a factor of P(x).

Polynomial Equations

The polynomial equations are those expressions which are made up of multiple constants and variables. The standard form of writing a polynomial equation is to put the highest degree first then, at last, the constant term. An example of a polynomial equation is:

b = a⁴ +3a³ -2a² +a +1

Polynomial Functions

A polynomial function is an expression constructed with one or more terms of variables with constant exponents. If there are real numbers denoted by a, then function with one variable and of degree n can be written as:

f(x) = a0xn + a1xn-1 + a2xn-2 + ….. + an-2x2 + an-1x + an

Solving Polynomials

Any polynomial can be easily solved using basic algebra and factorization concepts. While solving the polynomial equation, the first step is to set the right-hand side as 0. The explanation of a polynomial solution is explained in two different ways:

• Solving Linear Polynomials
• Solving Quadratic Polynomials

Solving Linear Polynomials

Getting the solution of linear polynomials is easy and simple. First, isolate the variable term and make the equation as equal to zero. Then solve as basic algebra operation. An example of finding the solution of a linear equation is given below:

Example: Solve 3x – 9

Solution:

First, make the equation as 0. So,

3x – 9 = 0

⇒ 3x = 9

⇒ x = 9/3

Or, x = 3.

Thus, the solution of 3x-9 is x = 3.

Solving Quadratic Polynomials

To solve a quadratic polynomial, first, rewrite the expression in the descending order of degree. Then, equate the equation and perform polynomial factorization to get the solution of the equation. An example to find the solution of a quadratic polynomial is given below for better understanding.

Example: Solve 3x² – 6x + x³ – 18

Solution:

First, arrange the polynomial in the descending order of degree and equate to zero.

⇒ x³ + 3x² -6x – 18 = 0

Now, take the common terms.

x²(x+3) – 6(x+3) =0

⇒ (x²-6)(x+3)=0

So, the solutions will be x =-3 and

x² = 6

Or, x = √6

For a polynomial, there could be some values of the variable for which the polynomial will be zero. These values are called zeros of a polynomial. Sometimes, they are also referred to as roots of the polynomials. In general, we find the zeros of quadratic equations, to get the solutions for the given equation.

The standard form of a polynomial in x is a_nxⁿ + a_n-1x^n-1 +….. + a₁x + a₀, where a_n, a_n-1, ….. , a₁, a₀ are constants, a_n ≠0 and n is a whole number. For example, algebraic expressions such as √x + x + 5, x² + 1/x² are not polynomials because all exponents of x in terms of the expressions are not whole numbers.

Zeros of a polynomial can be defined as the points where the polynomial becomes zero as a whole. A polynomial having value zero (0) is called zero polynomial. The degree of a polynomial is the highest power of the variable x.

• A polynomial of degree 1 is known as a linear polynomial.
  The standard form is ax + b, where a and b are real numbers and a≠0.
  2x + 3 is a linear polynomial.
• A polynomial of degree 2 is known as a quadratic polynomial.
  Standard form is ax² + bx + c, where a, b and c are real numbers and a ≠ 0
  x²+ 3x + 4 is an example for quadratic polynomial.
• Polynomial of degree 3 is known as a cubic polynomial.
  Standard form is ax³+ bx² + cx + d, where a, b, c and d are real numbers and a≠0.
  x³ + 4x + 2 is an example for cubic polynomial.

Similarly,

y⁶ + 3y⁴ + y is a polynomial in y of degree 6.

Formula

Consider, P(x) = 4x + 5 to be a linear polynomial in one variable.

Let ‘a’ be zero of P(x), then,

P(a) = 4k+5 = 0

Therefore, k = -5/4

In general, if k is zero of the linear polynomial in one variable: P(x) = ax +b, then;

P(k) = ak+b = 0

k = -b/a

It can also be written as,

Zero of Polynomial K = -(Constant/ Coefficient of x) 

Solved Example

Example 1: What is the value of ‘a’ if degree of polynomial, x³ + x^a-4 + x² + 1, is 4?

Solution:

Degree of a polynomial P(x) is the highest power of x in P(x).

Therefore, x^a-4  = x⁴

a-4 = 4, a = 4+4 =8

Therefore, the value of ‘a’ is 8.

Note: In general; if P(x) is a polynomial in x and k is any real number, then the value of P(k) at x = k is denoted by P(k) is found by replacing x by k in P(x).

Example 2: In the polynomial x² – 3x + 2,

Replacing x by 1 gives,

P(1) = 1 – 3 + 2 = 0

Similarly, replacing x by 2 gives,

P(2) = 4-6+2 = 0

For a polynomial P(x), real number k is said to be zero of polynomial P(x), if P(k) = 0.

Therefore, 1 and 2 are the zeros of polynomial x² – 3x + 2.