Polynomials

Polynomial is classified on the basis of its highest degree.

Degree of polynomial is nothing but the highest power of the variable in a polynomial.

e.g.- (ax + b) is in the form of linear polynomial.                    [Degree = 1]

        (ax2 + bx + c) is in the form of quadratic polynomial.     [Degree = 2]

        (ax3 + bx2 + cx + d) is in the form of cubic polynomial.  [Degree = 3]

        Where, a, b, c, d are real numbers and a ≠ 0.

Similarly, it can go on to more degrees, but we only need to study up to cubic polynomial.

Note     The degree of a polynomial has to be in the form of natural number (i.e.,       not in decimal or inverse/-ve) and if it is not then it is not a polynomial.

      e.g.-

are not polynomials.

The value of a polynomial can be calculated by substituting a numeric value of the variable.

e.g.- consider a polynomial p(x) = x2 – 3x – 4.

                   When x = 0;      p(0) = 0 – 0 – 4 = -4           

                   When x = 2;      p(2) = 4 – 12 – 4 = -12

                   When x = 4;      p(4) = 16 – 12 – 4 = 0       

                   When x = -1;     p(-1) = 1 + 3 – 4 = 0                                

 So, any value of x can be substituted to find the value of a polynomial.

 Generally, a real number k is said to be a zero of a polynomial p(x), if p(k) = 0.

Finding zero of a linear polynomial:

If k is a zero of p(x) = ax + b, then p(k) = ak + b = 0, i.e., k = (-b/a)

So, the zero of the linear polynomial ax + b is

                                     -b/a = -(constant term)/coefficient of x

Thus, the zero of a linear polynomial is related to its coefficients.

Relationship between zeroes and coefficient of a polynomia

  1. Zeroe of a linear polynomial ax + b is -b/a.
  2. Zeroes of a quadratic polynomial ax2 + b + c

    Let the zeroes be α & β; then,

                           α + β = –b/a

                               αβ = c/a

e.g.- p(x) = 2x2 – 8x + 6

                = 2x2 – (6+2)x + 6 = 2x2 – 6x – 2x + 6 = 2x(x – 3) – 2(x – 3)

P(x) = (x – 3)(2x – 2)

⸫ Zeroes of p(x) = 3 & 1         [BY equating p(x) = 0]

Now, we can have the relation of zeroes of a quadratic polynomial with its coefficient as:

                            α = 3  &  β = 1

                              α + β = 3 + 1 = 4 = –b/a = – (–8/2)

                               αβ = 3×1 = 3 = c/a = 6/2

Hence, Sum of zeroes = – (coefficient of x)/(coefficient of x2),

            Product of zeroes = constant term/coefficient of x2

  Also, a quadratic polynomial can also be written as:

                       X2 + (-sum of zeroes)x + (product of zeroes)

                       X2 + {-(α + β)}x + (αβ)

Example:

Find a quadratic polynomial, the sum and product of whose zeroes are –3 and 2, respectively.

Solution:

Let the zeroes be α & β, then

                                              α + β = –3

                                                  αβ = 2

 Using, X2 + {-(α + β)}x + (αβ)

Hence, the polynomial is x2 + 3x + 2.

For cubic polynomial, ax3 + bx2 + cx + d

α + β + γ  = –b/a,  (sum of zeroes)

αβ + βγ + γα = c/a, (sum of the product of zeroes taken two at a time)

αβγ = –d/a   (product of zeroes)

 Hence, a cubic polynomial can be written in the form:

      X3 + {-(α + β + γ)}x2 + (αβ + βγ + γα)x + (-αβγ)

Division Algorithm for polynomials

This is basically very important to find the other two zeroes of a cubic polynomial if one of its zero is given.

Let’s see in the example:

            X3 – 3x2 – x + 3

Now, by trial and error one of its zeroes can be determined as

          Put x = 1;      (1)3 – 3(1)2 – 1 + 3 = 0

                 ⸫   ‘1’ is one of the zero.

Hence, when the polynomial X3 – 3x2 – x + 3 is divided by (x – 1), then we get the quotient in the form of quadratic equation from which the other two zeroes can be calculated.

 Now, Factorising  x2 – 2x – 3

                              (x + 1)(x – 3) = 0        x = –1  &  3 ( the other two zeroes)

Categories: General

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