|Table of contents|
|Zeroes or roots of a polynomial|
|Relationship between Zeroes and Coefficients of a Polynomial|
|Identities of Algebra|
A polynomial is an algebraic expression in which the variables have non-integral exponents only.
Example- 3x2+ 4y + 2, 5x3+ 3x2 + 4x +2
For a polynomial in one variable - The highest exponent on the variable in a polynomial is the degree of the polynomial.
Example: The degree of the polynomial x2+3x+4 is 3, as the highest power of x in the given expression is x2.
It is that value of a variable at which polynomial P(x) becomes zero.
Example: if polynomial P(x) = x3-6x2+11x-6, Putting x = 1 one get P(1) = 0 then 1 is a zero of polynomial P(x).
The division algorithm states that for any given polynomial p(x) and any non-zero polynomial g(x) there are polynomial q(x) are r(x) such that
p(x) = g(x) × q(x) + r(x)
Dividend = Divisor × Quotient + Remainder
Where r(x) = 0 or degree r(x) < degree g(x)
A polynomial of degree n has at most n zeros.
Quadratic polynomials can be factorized by splitting the middle term.
For example, consider the polynomial 2x2−5x+3
Splitting the middle term
The middle term in the polynomial is -5x
Sum = -5
Product = 6
Now, -5 can be expressed as (-2) + (-3) and -2 x -3 = 6
Putting the above value in the gven expression
2x2−5x+3 = 2x2−2x−3x+3
Identify the common factor
2x2−2x−3x+3 = 2x(x−1)−3(x−1)
Taking (x−1) as the common factor, this can be expressed as:
(a+b)2 = a2+2ab+b2
(x+a)(x+b) = x2+(a+b)x+ab
a2−b2 = (a+b)(a−b)
a3−b3 = (a−b)(a2+ab+b2)
a3+b3 = (a+b)(a2−ab+b2)
(a+b)3 = a3+3a2b+3ab2+b3
(a−b)3 = a3−3a2b+3ab2−b3