Polynomials | Study

A polynomial in x can be denoted by the symbols p(x), q(x), f(x), g(x), etc.

Degree Of Polynomial: The highest power of x in p(x) is called the degree of the polynomial p(x).

Linear Polynomial : A polynomial of degree one is called a linear polynomial.

Quadratic Polynomial : 

A polynomial of degree two is called a Quadratic Polynomial.

Generally, any quadratic polynomial in x is of the form ax²+bx+c, a ≠ 0 and a, b, c are real numbers.

Cubic Polynomial : 

A polynomial of degree three is called a Cubic Polynomial.

Generally, any cubic polynomial in x is of the form ax³+bx²+cx+d, a≠0 and a, b, c, d are real numbers.

Value of a Polynomial : 

If we replace x by ‘ -2’ in the polynomial p(x) = 3x³-2x²+x-1

we have p(-2) =3(-2)³-2(-2)²+(-2)-1

= -24-8-2-1 =-35

Thus, on replacing x by ‘ -2 ‘ in the polynomial p(x), we get -35, which is called the value of the polynomial.

Hence, if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of the polynomial p(x) at x=k, and generally, denoted by p(k).

Zeros of a Polynomial : 

A real constant, k is said to be a zero of a polynomial p(x) in x, if p(k)=0

For example, the polynomial p(x) = x²+x-12 gives p(3)=3²+3-12=0 and  p(-4)=(-4)²+(-4)-12=0. 

Thus, 3 and -4 are two zeroes of the polynomial p(x).

A linear polynomial (degree one) has one and only one zero, given by;

Zero of the linear polynomial =  -(constant term )coefficient of  x

Geometrical Representation of the Zeroes of a Polynomial :

Let us consider a linear polynomial p(x)=3x-6.

We know that, graph of a linear polynomial is a straight line.

Therefore, graph of p(x)=3x-6 is a straight line passing through the points (1,-3),(3,3),(2,0).

Table for p(x)=3x – 6 

From the graph of p(x)=3x-6, we observe that it intersects the x-axis at the point (2,0). 

Zero of the polynomial [p(x)=3x-6] = -(-6)3 = 63 = 2.

Thus, we conclude that the zero of the polynomial p(x) = 3x – 6  is the x-coordinate of the point where the graph of p(x) = 3x – 6  intersects the x-axis.

Similarly, the zeroes of a quadratic polynomial, p(x) = ax²+bx+c, a≠0, are the x-coordinates of the points where the graph (parabola) of p(x)=ax²+bx+c, a≠0, intersects the x-axis.

Graph of p(x) = ax²+bx+c, a≠0 intersects the x-axis at the most in two points and hence the quadratic polynomial can have at most two distinct real zeros.

A cubic polynomial can have at most three distinct real zeros.

Relation between Zeroes and Coefficients of a Polynomial : 

Let the quadratic polynomial be p(x) = ax²+bx+c, a≠0 and having zeroes as α and β, then

Sum of the zeroes = α + β

= -(coefficient of  x) /(coefficient of  x²) = -b/a

Product of the zeroes = αβ

Let the cubic polynomial be p(x) = ax³+bx²+cx+d, a≠0 and having zeroes as α , β and γ, then Sum of the zeroes = α + β + γ

α + β + γ = -(coefficient of  x² )/(coefficient of  x³)= -b/a

αβ = (constant term) /(coefficient of  x²) = c/a

Sum of the products of zeroes taken two at a time αβ+βγ+γα

αβ+βγ+γα = (coefficient of  x) /(coefficient of  x³)= c/a

and

Product of the zeroes = αβγ

αβγ = (constant term) /(coefficient of  x³)= -d/a

Division Algorithm for Polynomials : 

For any two polynomials p(x) and g(x) ; g(x) ≠0, we can find two 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) is less than the degree of g(x). Here, q(x) is called quotient, r(x) is called remainder, p(x) is called dividend and g(x) is called divisor. This result is known as a division algorithm for polynomials.