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 nonnegative 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} ≠ 0
2. 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 coefficients of x^{0}, x^{1}, x^{2}, ....., x^{n1}, x^{n} respectively.
The highest power of the variable in a polynomial is called its degree.
Example: 5x + 3 is a polynomial in x of degree 1.
p(y) = 3y^{2} + 4y  4 is a polynomial in y of degree 2.
Examples:
If p (x) is a polynomial in x and ‘a’ 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.
A real number ‘a’ 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
∴ (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.
First, we consider a linear polynomial p (x) = ax + b.
Let ‘k’ be a zero, then p(k) = ak + b = 0
⇒ ak + b = 0
⇒ ak =  b or
The graph of a linear polynomial is always a straight line. It may or may not pass through the xaxis. In case the graph line is passing through a point on the xaxis, then the ycoordinate of that point must be zero. In general, for a linear polynomial ax + b = 0, (a ≠ 0), the graph is a straight line that can intersect the xaxis at exactly one point, namely, is the zero of the polynomial ax + b.
In the given figure, CD is meeting xaxis at x = 1.
∴ Zero of ax + b is 1.
Note:
A zero of a linear polynomial is the xcoordinate of the point, where the graph intersects the xaxis.
The graph of ax^{2} + bx + c, (a ≠0) is a curve of ∪ shape, called a parabola.
In the given figure, the graph of a quadratic polynomial x^{2}  3x  4 is shown. It intersects xaxis 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 xaxis 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.
Note: In the case of Quadratic polynomial  at most 2 zeroes, Cubic polynomial  at most 3 zeroes, Biquadratic polynomial  at most 4 zeroes.
To divide one polynomial by another, follow the steps given below.
Step 1: Arrange the terms of the dividend and the divisor in the decreasing order of their degrees.
Step 2: To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor Then carry out the division process.
Step 3: The remainder from the previous division becomes the dividend for the next step. Repeat this process until the degree of the remainder is less than the degree of the divisor.
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