# Polynomial

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

Polynomial of degree 1 is known as linear polynomial.
Standard form is ax+b, where a and b are real numbers and a ≠ 0.
4x+1 is a linear polynomial.

Polynomial of degree 2 is known as quadratic polynomial.
Standard form is ax2 + bx + c, where a, b and c are real numbers and a ≠ 0
x2 + 4x + 6 is an example for quadratic polynomial.

Polynomial of degree 3 is known as cubic polynomial.
Standard form is ax3 + bx2 + cx + d, where a, b, c and d are real numbers and a ≠ 0.
x3 + 2x + 6 is an example for cubic polynomial.

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

p(x) = x3 + 4x + 5
at x = -1 we have
p(-1) = (-1)3 + 4(-1) + 5
= -1 - 4 + 5
= 0
As p(–1) = 0, –1 is called the zeroes of the quadratic polynomial x3 + 4x + 5. More generally, a real number k is said to be a zero of a polynomial p(x), if p(k) = 0 the zeroes of a quadratic polynomial ax2 + bx + c, a ≠ 0, are precisely the x-coordinates of the points where the parabola representing y = ax2 + bx + c intersects the x-axis.

In general, given a polynomial p(x) of degree n, the graph of y = p(x) intersects the x- axis at atmost n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.

If α and β are zeros of the polynomial then
Sum of its zeroes α + β
= -(Coefficient of x)/(Coefficient of x2) = -b/a

Product of zeroes = αβ
Constant term/Coefficient of x2 = c/a

If α, β, γ are the zeroes of the cubic polynomial ax3 + bx2 + cx + d, then

α + β + γ = -b/a
αβ + αγ + βγ = c/a
αβγ = -d/a

Division Algorithm
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).