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**Facts that Matter**

**Polynomial**

An algebraic expression of the form p(x) = a_{0}+ a_{1}x + a_{2}x^{2}+ a_{3}x^{3}+ ...... + a_{n}x^{n}, in which the variables involved have only non-negative integral exponents, is called a polynomial in x of degree n.

Note:In the polynomial a_{0}+ a_{1}x + a_{2}x^{2}+ .... + a_{n}x^{n}1. a_{n}â‰ 02. a_{0}, a_{1}x^{1}, a_{2}x^{2}, a_{3}x^{3}, ..... a_{n - 1}x^{n - 1}, a_{n}x^{n}are terms.3. a_{0}, a_{1}, a_{2}, ..... a_{n - 1}, a_{n}are the co-efficients of x^{0}, x^{1}, x^{2}, ....., x^{n-1}, x^{n}respectively.

**Degree of a Polynomial**

The highest power of the variable in a polynomial is called its degree. For example,

p(x) = 5x + 3 is a polynomial in x of degree 1.

p(y) = 3y^{2}+ 4y - 4 is a polynomial in y of degree 2.**Linear Polynomial**

A polynomial of degree 1 is called a linear polynomial. A linear polynomial is generally written as ax + b (a â‰ 0), where a, b are real coefficients.**Quadratic Polynomial**

A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial is generally written as ax^{2}+ bx + c (a â‰ 0), where a, b and c are real coefficients.**Cubic Polynomial**

A polynomial of degree 3 is called a cubic polynomial. A cubic polynomial is generally written as ax^{3}+ bx^{2}+ cx + d (a â‰ 0), where a, b, c and d are real coefficients.**Examples:****Degree of different polynomials**

**Value of a Polynomial at a Given Point**

If p (x) is a polynomial in x and â€˜aâ€™ is a real number. Then the value obtained by putting

x = a in p (x) is called the value of p (x) at x = a.

Example:

Let p(x) = 5x^{2}- 4x + 2 then its value at x = 2 is given by

p (2) = 5 (2)^{2}- 4 (2) + 2

= 5 (4) - 8 + 2

= 20 - 8 + 2

= 14

Thus, the value of p(x) at x = 2 is 14.**Zeroes of a Polynomial**

A real number â€˜aâ€™ is said to be a zero of the polynomial p (x), if p (a) = 0.

Example:

Let p (x) = x^{2}- x - 2

Then p (2) = (2)^{2}- (2) - 2 = 4 - 4 = 0

and p (-1) = (-1)^{2}- (-1) - 2 = 2 - 2 = 0

âˆ´ (-1) and (2) are the zeroes of the polynomial x^{2 }- x - 2.

Note:I. A linear polynomial has at the most one zero.

II. A quadratic polynomial has at the most two zeroes.III. In general a polynomial of degree n has at the most n zeroes.

**Geometrical Meaning of the Zeroes of a Polynomial**

First, we consider a linear polynomial p (x) = ax + b. Let â€˜kâ€™ be a zero, then

p (k) = ak + b = 0

â‡’ ak + b = 0

â‡’ ak = - b

or k =

The graph of a linear polynomial is always a straight line. It may or may not pass through

x-axis. In case the graph line is passing through a point on the x-axis, then the y-coordinate of that point must be zero.

In general, for a linear polynomial ax + b = 0, (a â‰ 0), the graph is a straight line which can intersect the x-axis at exactly one point, namely, is the zero of the polynomial ax + b.In the given figure, CD is meeting x-axis at x = -1.

âˆ´ Zero of ax + b is -1.

Note:A zero of a linear polynomial is the x-coordinate of the point, where the graph intersects the x-axis.

**Graph of a Quadratic Polynomial**

The graph of ax^{2}+ bx + c, (a â‰ 0) is a curve of âˆª shape, called a parabola. If a > 0 in

ax^{2}+ bx + c, the shape of the parabola is âˆª (opening upwards). If a < 0 in ax^{2}+ bx + c, the shape of parabola is âˆ© (opening downwards).

In the given figure, the graph of a quadratic polynomial x^{2}- 3x - 4 is shown. It intersects x-axis at (-1, 0) and (4, 0). Therefore, its zeroes are -1 and 4. Here, a > 0, so the graph opens upwards.

Whereas the following figure is a graph of the polynomial - x^{2}+ x + 6. Since it intersects the x-axis at (3, 0) and (-2, 0). Therefore, the zeroes of - x^{2}+ x + 6 are -2 and 3.

Here a < 0, so the parabola opens downwards.

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