Polynomials
Definition
where a_0,a_1,a_2, . . . ,a_n are constants and a_n is not equal to 0
Terms
a polynomial of 2 terms is called a binomial.
A polynomial of three terms is called a trinomial
A polynomial of 1 term is called a monomial.
A polynomial p(x) in one variable x is an algebraic expressionin x of the form
p(x) = *a_nx^n + a_n-1x^n-1 + . . . + a_2x^2 + a_1x + a_0
Degrees
A polynomial of 1 degree is called linear polynomial.
A polynomial of 2 degrees is called as quadratic polynomial
A polynomial of 3 degrees is called a cubic polynomial
A real number a is a zero of a polynomial p(x) if p(a) = 0.
Every linear polynomial in one variablehas a unique 0
Identities
(a + b) 2 = a 2 + b 2 + 2ab.
(a − b) 2 = a 2 + b 2 − 2ab
a 2 − b 2 = (a − b) (a + b)
(x + a) (x + b) = x 2 + (a + b) x + ab.
(x + y)3 = x3 + y3 + 3xy(x + y)
(x – y)3 = x3 – y3 – 3xy(x – y)
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)
Remainder theorem
Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a). ... Solution: Zero of x – 1 is 1, so as per remainder theorem remainder in this case will be p(1) .
Factor Theorem
In mathematics, factor theorem is used when factoring the polynomials completely. It is a theorem that links factors and zeros of the polynomial. According to factor theorem, if f(x) is a polynomial of degree n ≥ 1 and 'a' is any real number, then, (x-a) is a factor of f(x), if f(a)=0.
Coefficients
coefficient is an integer that is written along with a variable or it is multiplied by the variable. In other words, a coefficient is the numerical factor of a term containing constant and variables. For example, in the term 2x, 2 is the coefficient.