Well studied algebraic expressions, their addition, subtraction, multiplication and division in earlier classes.
(x + y)2 = x2 + 2xy + y2
(x — y)2 = x2 —2xy + y2
Here we start with a particular type of algebraic expression, called polynomial, and there terminology.
We learnt that a variable is denoted by a symbol that can take any real value it can be any alphabets.
There is a difference between a letter denoting a constant and a letter denoting a variable. The values of the constants remain the same throughout a particular situation, that is, the values of the constants do not change in a given problem, but the value of a variable can keep changing.
We know much about the variables use in algebraic expression like - a, x, y etc.
Till now we have leant all the algebraic expressions we have considered so far have only whole numbers as the exponents of the variable.
As we know about the polynomials that has more than one variables. 3y2 + 5y is a polynomial in the variable y and t2 + 4 is a polynomial in the variable t.
In the expressions y2 and 5y are called the terms of the polynomial.
Each term of a polynomial has a coefficient. So, in —x3 + 4x2 + 7x —2, the coefficient of x3 is —1, the coefficient of x2 is 4, the coefficient of x is 7 and —2 is the coefficient of x0 (Remember, x0 = 1).
The constant polynomial 0 is called the zero polynomial.
A polynomial can have any (finite) number of terms.
Polynomials having only two terms are called binomials (‘bi’ means ‘two’). Similarly, polynomials having only three terms are called trinomials (‘tri’ means ‘three’).
The highest power (exponent) of the variable in a polynomial as the degree of the polynomial.
A polynomial of degree one is called a linear polynomial.
A polynomial of degree two is called a quadratic polynomial.
A polynomial of degree three a cubic polynomial.
If p(1) = 0, we say that 1 is a zero of the polynomial p(x).
A non-zero constant polynomial has no zero.
Every real number is a zero of the zero polynomial.
The dividend becomes the remainder and the sum of the quotients gives us the whole quotient.
Dividend = (Divisor x Quotient) + Remainder
The factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers.
Following algebraic identities in earlier classes:
Identity I : (x + y)2 = x2 + 2xy + y2
Identity II : (x — y)2 = x2 — 2xy + y2
Identity III : x2 — y2 = (x + y) (x — y)
Identity IV : (x + a) (x + b) = x2 + (a + b)x + ab
Identity V : (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
Identity VI : (x + y)3 = x3 + y3 + 3xy (x + y)
Identity VII : (x — y)3 = x3 — y3 — 3xy(x — y)
= x3 — 3x2y + 3xy2 — y3
Identity VIII : x3 + y3 + z3 — 3xyz = (x + y + z)(x2 + y2 + z2 — xy — yz — zx)