**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]

(ax^{2} + bx + c) is in the form of quadratic polynomial. [Degree = 2]

(ax^{3} + bx^{2} + 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) = x^{2} – 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**

- Zeroe of a linear polynomial ax + b is -b/a.
- Zeroes of a quadratic polynomial ax
^{2}+ b + c

Let the zeroes be **α** & **β**; then,

** α + β = –b/a**

** αβ = c/a**

e.g.- p(x) = 2x^{2} – 8x + 6

= 2x^{2} – (6+2)x + 6 = 2x^{2} – 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 x^{2}),

Product of zeroes = constant term/coefficient of x^{2}

^{ }Also, a quadratic polynomial can also be written as:

X^{2} + (-sum of zeroes)x + (product of zeroes)

X^{2} + {-(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, X^{2} + {-(α + β)}x + (αβ)

Hence, the polynomial is x^{2} + 3x + 2.

**For cubic polynomial**, ax^{3} + bx^{2} + 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:

X^{3} + {-(α + β + γ)}x^{2} + (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:

X^{3} – 3x^{2} – 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 X^{3} – 3x^{2} – 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 x^{2} – 2x – 3

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

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