Polynomials

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Sub Topics Polynomial Equation Polynomial Functions Types of Polynomials Irreducible Polynomials Factoring Polynomials Roots of Polynomials Graphing Polynomial Functions Operations with Polynomials Polynomial Fractions Polynomial Word Problems Polynomial Examples

The word polynomial literally means "many terms". Polynomials are made up of constants and variables connected with algebraic operations, whereas the exponent of a variable must be a non-negative integer. For example, $x+a$, $x^{2}-a^{2}+bx+c$, $x^{3}+3x^{2}+2x+1$, $z^{3}-y^{3}-7yz-6$ etc. are polynomials. On the other hand, $y^{-2}+y-1$, $x+\frac{1}{x}$ and $x^{2}-x$$^{\frac{2}{3}}$$+1$ are not polynomials, since the exponents of variables are not non-negative integers. The degree of a polynomial is the greatest exponent of a variable. The degree of polynomial $2x+5$ is 1. The degree of polynomial $3x^{2}+7x-6$ is 2. The degree of $6x^{5}-3x^{2}y^{3}+7xy^{2}$ is 5. The general form of the polynomial is Above polynomial is known as polynomial of degree n, provided that $a_{n}\neq 0$ and every exponent of x is a non-negative integer.

An equation containing polynomial is known as polynomial equation. In other words, when a polynomial is represented in the form of equation, it is known as polynomial equation. For example, $x^{2}+3x-2$ is a polynomial, while $x^{2}+3x-2=0$ is a polynomial equation.
A function which contains polynomial is called polynomial function. So, we can write that function of the form:
Above polynomial is known as polynomial function.
For example, $f(x)=x^{3}+x^{2}y^{2}+y^{3}$ is a polynomial function.
Polynomials are classified into various types. There are two categories on the basis of which polynomials are classified.

Classification on the Basis of Degree:

• Linear Polynomial: A polynomial of degree "one" is known as linear polynomial such as $3x+5$.
• Quadratic Polynomial: A polynomial of degree "two" is known as quadratic polynomial. For example, $3x^{2}-7x-8$.
• Cubic Polynomial: A polynomial of degree "three" is known as cubic polynomial such as $x^{3}+2x^{2}-x+2$.
• Bi-quadratic Polynomial: A polynomial of degree "four" is known as bi-quadratic polynomial, like $x^{4}-3x^{3}+2x-3$.
  

Higher degree polynomials are also named under this category. But, above four types of polynomials are used most often.

Classification on the Basis of Number of Terms:

• Monomial: A polynomial containing one non-zero term is called monomial. For example, 5, 3x etc.
• Binomial: A polynomial containing two non-zero terms is called binomial. For example, 3 + 6x, x - 5y etc.
• Trinomial: A polynomial containing three non-zero terms. For example, $x^{5}-x^{2}y^{3}+y^{5}$, $x^{2}-7x+9$ etc.

Except above classification, we have zero polynomial and constant polynomial too.

• Zero Polynomial: A polynomial containing only one term that is zero is called zero polynomial.
• Constant Polynomial: A polynomial containing only one constant term is called constant polynomial. For example, 7, 28, -4 etc.

Polynomial can be factorized down and written in the form of product of lower degree polynomials. For example, A polynomial $x^{2}+3x+2$ can be factorized and expressed as follows:
$x^{2}+3x+2=(x+1)(x+2)$.
Such polynomials are known as reducible polynomial.
In contrast with reducible polynomials, irreducible polynomial are those which cannot (by any means) be factorized. For example, $x^{2}+x+1$ and $x^{2}+1$ are irreducible polynomials. Such polynomials are also known as prime polynomials.
Polynomials are factorized by different methods. Few important methods are listed below:

1. Finding common factor: Polynomials can be factorized by finding common terms.
Consider $x^{2}+2x$
Here, x is common in both the terms. So, we can write it in the following way:
$x^{2}+2x$ = $x(x+2)$

2. Grouping and Finding Common Factor: Sometimes, we need to group like terms and then take the common factor out.
Consider $x^{3}+x^{2}y+y^{3}+xy^{2}$
$x^{3}+x^{2}y+y^{3}+xy^{2}$ = $(x^{3}+x^{2}y)+(y^{3}+xy^{2})$
= $x^{2}(x+y)+y^{2}(y+x)$
= $x^{2}(x+y)+y^{2}(x+y)$
= $(x+y)(x^{2}+y^{2})$

3. Decomposition Method: This method is used to factorize a quadratic polynomial. A quadratic polynomial of the form $ax^{2}+bx+c$ is to be factorized in the following way:

• Multiply the values of a and c (along with their signs).
• Determine the value of b (along with the sign).
• Split the product ac in two terms so that, after adding or subtracting those splits parts, we must get the product ac.
• Write those split parts in place of b.
• Factorize by grouping and taking common factor out.
  

4. Using Formulas: There are different formulas for factorizing polynomials:

• $a^{2}-b^{2}=(a+b)(a-b)$
• $(a+b)^{2}=a^{2}+2ab+b^{2}$
• $(a-b)^{2}=a^{2}-2ab+b^{2}$
• $(a+b)^{3}=a^{3}+b^{3}+3ab(a+b)$
• $(a-b)^{3}=a^{3}-b^{3}-3ab(a-b)$
• $a^{3}+b^{3}=(a+b)(a^{2}+b^{2}-ab)$
• $a^{3}-b^{3}=(a-b)(a^{2}+b^{2}+ab)$
• $(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ac$
• $a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ac)$

5. Completing Perfect Square: Another way to factorize a polynomial is to complete the perfect square. After completing perfect square, we may use formulas described above as per the requirement of question.

Consider $x^{2}-6x+5$
$x^{2}-6x+5$ = $x^{2}-6x+9-9+5$
= $(x^{2}-6x+9)-4$
= $(x-3)^{2}-2^{2}$

Using the identity, $a^{2}-b^{2}=(a+b)(a-b)$, we get
$(x-3+2)(x-3-2)$
$(x-1)(x-5)$
Roots of a polynomials are also known as zeros of polynomials. Roots or zeros of a polynomial are the points at which the value of a polynomial is zero. In other words,
The number "a" is said to be the root or zero of a polynomial P(x), if P(a) = 0
To find the roots or zeros of a polynomial, we simply need to equate the polynomial to zero and determine the values of variable contained in it.
The number of roots or zeros of a polynomial are equal to its degree.

Consider $P(x)=x^{2}-3x$
Since the degree of given polynomial is 2, hence it must have 2 roots.

To find the zeros of P(x), we take P(x) = 0
$x^{2}-3x=0$
$x(x-3)=0$
$x=0\ and\ x=3$
Therefore, 0 and 3 are two zeros of polynomial P(x).
The graph of a polynomial function,$a_nx^n + a_{n-1}x^{n-1} + ....+ a_1x + a_0$ depends upon two factors:

• Whether 'a' is positive or negative.
• Whether 'n' is even or odd.

The end behavior of a polynomial function graph is depending upon following four cases:
Case 1: $a> 0$ and n is even
In this case, graph rises at both left and right endpoints.


Case 2: $a> 0$ and n is odd
In this case, graph falls at left endpoint and rises at right endpoint.

Case 3: $a< 0$ and n is even
In this case, graph falls at both left and right endpoints.

Case 4: $a< 0$ and n is odd
In this case, graph rises at left endpoint and falls at right endpoint.

Basic algebraic operations on polynomials are performed in the following manner:
1. Addition: Combine like terms and then proceed.
2. Subtraction: Flip the signs of the terms of polynomial which is to be subtracted, then combine like terms and proceed.
3. Multiplication: While multiplying, following rules of exponents should be kept in mind:

• $a^{m} \times a^{n}=a^{mn}$
• $a^{m} \times b^{m}=(ab)^{m}$
• $(a^{m})^{n}=a^{mn}$

Along with above rules, distributive law is also remembered: $a(b+c)=ab+ac$
4. Division: While dividing, divide each term separately and follow the following rules of exponents:

• $\frac{a^{m}}{a^{n}}$ = $(a)$$^{m-n}$
• $\frac{a^{m}}{b^{m}}=(\frac{a}{b})^{m}$
• $\frac{1}{a^{m}}$ = $a$$^{-m}$

To simplify polynomial fraction, we should first factorize both numerator and denominator polynomials and then cancel the common factors if there is any.

Let us simplify $\frac{x^{2}-4}{x+2}$

$\frac{x^{2}-4}{x+2}$ = $\frac{(x+2)(x-2)}{x+2}$

$(x+2)$ in numerator is cancelled out from $(x+2)$ in denominator and we are left with $(x-2)$

=> $\frac{x^{2}-4}{x+2}$ = x - 2
Few word problems based on polynomials are given below: Few examples based on polynomials are shown below: