Introduction of Polynomials

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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