.Chapter 2
Introduction to Polynomials
Polynomial: A polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents.
Let x be the variable, n be a positive integer and a1, a2, a3, …, an be the real numbers, then
p(x) = anxn + an-1xn-1 + an-2xn-2 + … + a1x + a0 is called a polynomial in x.
Degree of polynomial: The highest power of x in a polynomial p(x) is called the degree of the polynomial p(x).
Linear polynomial: A polynomial of degree 1 is called a linear polynomial.
Quadratic polynomial: A quadratic polynomial is a polynomial whose degree is 2.
Cubic polynomial: A cubic polynomial is a polynomial whose degree is 3.
If p (x) is a polynomial in x, and if m is any real number, then the value obtained by replacing x by m in p(x), is called the value of p(x) at x = m.
Zero of polynomial: A real number m is said to be a zero of a polynomial p(x), if p(m) = 0.
In
general, if m
is
a zero of p(x)
= ax
+
b,
then p(m)
= am
+
b
=
0, i.e., m =.
So,
the zero of the linear polynomial ax
+
b
is
.
Geometrical representation of polynomial
(a)
The graph of a linear polynomial is a straight line which intersects
the x-axis
at exactly one point, namely
(b) For any quadratic polynomial ax2 + bx + c, a ≠ 0, the graph of the corresponding equation y = ax2 + bx + c has one of the two shapes either open upwards parabola or open downwards parabola depending on whether a > 0 or a < 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.
(c) 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.
Relationship between Zeroes and Coefficients of a Polynomial
(1) In general, if and are the zeroes of the quadratic polynomial
p (x) = ax2 + bx + c, a ≠ 0, then,
Sum
of its zeroes
=
+
=
Product
of its zeroes
=
=
(2) In general, it can be proved that if , , are the zeroes of the cubic polynomial
ax3 + bx2 + cx + d, then
Sum
of its zeroes
=
+
+
=
Product
of zeroes taken any two at time
=
+
+
=
Product
of zeroes
=
=
Division algorithm
“Dividend = Divisor × Quotient + Remainder”
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)
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