What We Have Discussed

A polynomial p(x) in one variable x is an algebraic expression in x of the form: p(x) = anxn + an−1xn−1 + ....... + a2x2 + a1x + a₍₀₎ where a0 , a1 , a2 , ... an are respectively the coefficients of x0, x1, x2, ....xn and n is called the degree of the polynomial if an ≠ 0. Each anxn, an−1xn−1, .... a0 , is called a of the polynomial p(x).

Polynomials are classified as monomial, binomial, trinomial etc. according to the number of in it.

Polynomials are also named as linear polynomial, quadratic polynomial, cubic polynomial etc. according to the of the polynomial.

A real number 'a' is a of a polynomial p(x) if p(a) = 0. In this case, 'a' is also called a root of the polynomial equation p(x) = 0.

Every linear polynomial in one variable has a zero, a non-zero constant polynomial has no zero.

Remainder Theorem : If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial (x - a), then the remainder is .

Factor Theorem : If x - a is a factor of the polynomial p(x), then p(a) = 0. Also if p(a) = 0 then (x-a)(x+a) is a factor of p(x).

Some Algebraic Identities:

(i) x+y+z2 ≡ x2+y2+z2+2xy+2yz+2zx

(ii) x+y3 ≡ x3+y3+3xyx+y

(iii) x−y3 ≡ x3−y3−3xyx−y

(iv) x3+y3+z3−3xyz ≡ x+y+zx2+y2+z2−xy−yz−zx

(v) x3+y3 ≡ (x + y)x2−xy+y2

(vi) x3−y3 ≡ (x - y)x2+xy+y2

(vii) x4+4y4 ≡ x+y2+y2x−y2+y2