Introduction of Polynomials

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Polynomials are algebraic expressions that include real numbers and variables. Division and square roots cannot be involved in the variables. The variables can only include addition, subtraction and multiplication.

Let x be a variable, n be a positive integer and a
0 , a 1 ,a 2 ... ,a n be constants. Then,
f(x) = a
n x n + a n-1 x n-1 + ... + a 1 x + a 0 is called a Poly-nomial in variable x.

Here, a
n x n , a n-1 x n-1 , ... , a 1 x, a 0 are known as terms
and a
n , a n-1 , ..., a 1 ,a 0 are coefficients .

Examples

1) p(x) = 3x -2 -----> polynomial in variable x.

2) q(y) = 3y
2 - 2y + 4 ------> is a poly-nomial in y.

3) f(u)= 1/2 u
3 - 3u 2 + 2u - 4 ----> is a poly-nomial in variable u.

Not a Poly-nomial : If the exponent of any term is less than 1 or negative.

Examples :

1) 2x
2 - 3√x + 5

Solution :
It is not a poly-nomial because the exponent of one term is 1/2 which is less than 1.

2) 1/(x
2 - 2x + 5)

Solution :
It is not a poly-nomial because the exponent is negative.

3) 2x
3 - 3/x + 4

Solution :
As the exponent of 2nd term is -1 so it is not a poly-nomial.
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Practice

Q.1 Write any poly-nomial in x with degree 3.
Q.2 P(x) = x
3 + 2 3 - 4x + 8. Write the coefficient of x 2 .
Q.3 In P(x) = 5x
2 + x 5 - 2x + 4 write the degree of poly-nomial.
Q.4 In P(x) = 1/2 x
3 - 7x 2 + 8x, state the number of terms and write each terms.
Polynomials

Degree of the Poly-nomial
Zeros of Poly-nomial
Remainder Theorem
Find remainder by Synthetic Division
Rational root test in Poly-nomial
Solved Examples on Factorization

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