32. Polynomial, their operations and factors of polynomial.
32. Polynomial, their operations and factors of polynomial. |
Learn and Remember:
1) Polynomial :
(1) If a0, a1, a2, ..., an-1, an, are real numbers and n is a non-negative integer, then
a0+a1x+a2x2 +.............+an x n is called a polynomial in x.
e.g. 3x, 4x2y, a2+ 2ab+b2 , - 5m3 +3m2 +6, etc. are polynomials.
(ii) 3x-2 is not a polynomial, because the power of variable x is a negative integer.
(iii) The degree of a polynomial is a non-negative integer.
(iv) Standard form of a polynomial : If the terms of a polynomial are either in the ascending powers of the variable or in the descending power, then such a form is called the standard form of the polynomial.
2. The degree of a polynomial : The highest index of the variable in a given polynomial is called the degree of the polynomial.
e.g. the degree of the polynomial (1) x3 + 7x2 + 5x – 9 is 3.
(2) 27 = 27xº the degree of 27 is 0. •
♦ The degree of a non-zero constant monomial is zero.
♦ The degree of 0 is not defined.
3. Classification of Polynomials :
(1) Monomial : A polynomial having only one term is called a monomial.
e.g. 5x; 37; – √5x.
(2) Binomial : A polynomial having two terms is called a binomial.
e.g. 3x – 7; 5+n, x3 +4.
(3) Trinomial : A polynomial having three terms is called a trinomial.
e.g. 5x2 – 3x + 8; 15+ n3+ n5
♦ The method of addition, subtraction, multiplication and division is similar to the operations of algebraic expressions.
♦ The degree of the product of two polynomials is equal to the degree of multiplicand + the degree of multiplier.
e.g. the degree of the product of x4 +5 and x2+ 3 is 4 + 2 =6.
♦ Coefficient form of the polynomial : 2x4+ 3x2 +5
is a polynomial. We write it with the missing terms.2x4 + 0x3+ 3x2 + 0x +5. This polynomial is represented as (2,0,3, 0, 5). This form of the polynomial is called coefficient form of the polynomial. •
The degree of division of the polynomials is equal to the degree of the dividend - the degree of the divisor.
- Remainder theorem : If p(x), a polynomial of degree ≥ 1, is divided by (x - a), then the remainder is p((a).
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