Polynomials

Polynomials

A polynomial is an algebraic expression in form of 

where a₀, a₁, a₂,...a_n are real numbers and all  exponents of x are whole numbers.

is called polynomial in variable x and expressed as p(x).

Example:- x² + 17x + 23 is a polynomial in variable x.

Polynomial may contain one or more terms(finite terms) which are linked by mathematical operators (+,-,/,*). It may have one or more variables.

 x² + 17x + 23 has three terms, First term = x² , Second term = 17x, Third term = 23.

If you add or subtract or multiply polynomials the result is  another polynomial. 

Conditions 

1. It must contain no square roots of variables
2. No Fractional or negative powers on the variables, 
3. No variables in the denominators of any fractions.

Degree of polynomial

Polynomial containing one variable:

The largest exponent in any term of the polynomial is known degree of polynomial.

Polynomial containing  more than one variable:

Maximum value of sum of exponents of all variables in any term of polynomial is the degree of polynomial.

Leading term of polynomial

The term containing maximum exponent  (highest degree )on variable is called leading term.

Polynomials are commonly written with their terms in descending order of degree.

Types of polynomials

Liner polynomial:

A polynomial of degree 1 is termed  liner polynomial. 

General form of linear polynomial, p(x) = ax + b ,Where a& b are real number and a ≠0.

Linear polynomial is the simplest form of polynomials.

The graph of linear polynomial is a straight line.

Ex: p(y)= 4y + 5 is an example of linear polynomial.

Quadratic polynomial:

A polynomial of degree 2 is known as quadratic polynomial. 

General form quadratic polynomial p(x)=ax² + bx +c  where a, b& c are real number and a ≠0.

The graph of quadratic polynomial is a parabola.

Ex: p(x)= 4x² + 5x+3 is an example of quadratic polynomial.

Cubic polynomial:

A polynomial of degree 3 is called cubic polynomial. 

General form of cubic polynomial p(x)= ax³ + bx² + cx + d, a ≠ 0, where a, b, and c are coefficients and d is the constant with all of them being real numbers.

Ex: p(t): t³ − 5t² + 15t − 6 is an example of quadratic polynomial.

Zero of a polynomial

Zeroes of a polynomial p(y) is a number k such that p(k) = 0.It is determined by replacing y=k in polynomial(y).

Zero of a polynomial is also called the root of the polynomial.

The maximum number of zeroes of a polynomial is equal to its degree.

Relationship between co-efficient of polynomials and zeroes

Liner polynomial:

For polynomial p(x)= ax+b

Zero of a polynomial = -b/a

 

Zero of a polynomial = –(constant term/coefficient of x)

Quadratic polynomial:

The quadratic polynomial P(x) = ax2 + bx + c. Where, a ≠ 0. 

Let say α and β are the two zeros of polynomial p(x), then 

Sum of zeros( α + β ) = -b/a = – Coefficient of x/ Coefficient of x2

Product of zeros(αβ) = c/a = Constant term / Coefficient of x2

Cubic polynomial:

The cubic polynomial p(x) = ax3 + bx2 + cx + d, where a ≠ 0. 

Let α, β, and γ are the three zeros of a polynomial, 

Sum of zeros (α + β + γ) = -b/a = – Coefficient of x2/ coefficient of x3

Sum of the product of zeros taken two at a time (αβ+ βγ + αγ)

=>c/a = Coefficient of x/Coefficient of x3

Product of zeros(αβγ)= -d/a = – Constant term/Coefficient of x3

Remainder Theorem

Remainder theorem says that when a polynomial p(t) is divided by a linear polynomial (t - a), The remainder is equal to p(a).

Factor Theorem

Let p(x) be a polynomial of degree n ≥ 1 and 'a' be a real number such that p(a) = 0, then (x -a) is a factor of p(x). Conversely, if (x-a) is a factor of p(x), then p(a)=0

Division Algorithm 

Division algorithm for polynomials describes that, if p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x) × q(x) + r(x) 

where r(x) = 0 or degree of r(x) < degree of g(x). Here,

p(x)= dividend polynomial

g(x) = divisor polynomial

q(x) = quotient polynomial

r(x) = remainder polynomial

# Example

Go through the below-provided example to understand the division algorithm for polynomials, which is given in step by step procedure.

# Graph of polynomials