## Polynomials

### Polynomials in One Variable

#### Polynomial

The algebraic expression of the form p(x) = an xn + an-1 xn-1 +........+ a1 x1 + a0, where an, an-1, a1, a0 are real numbers and an≠ 0 is called a polynomial. The exponent of each term of a polynomial is a non-negative integer.

Monomial
A polynomial with only one term is called a monomial.

#### Binomial

A polynomial with only two terms is called a binomial.

#### Trinomial

A polynomial with only three terms is called a trinomial.

#### Zero Polynomial

The polynomial with all the coefficients as zeros is called a zero polynomial.

#### Constant Polynomial

A polynomial with a single term of a real number is called a constant polynomial.

#### Degree of a Polynomial

The exponent of the term with the highest power is called the degree of the polynomial. The degree of a non-zero constant polynomial is zero. The degree of the zero polynomial is not defined.

A polynomial is named according to the degree of the polynomial.

#### Linear Polynomial

A polynomial of degree one is called a first-degree or linear polynomial. The general form of a linear polynomial is ax + b, where a ≠ 0. Maximum number of terms of a linear polynomial is two.

A polynomial of degree two is called a second degree or quadratic polynomial. The general form of a quadratic polynomial is ax2 + bx + c, where a ≠ 0. Maximum number of terms of a quadratic polynomial is three.

#### Cubic Polynomial

A polynomial of degree three is called a third-degree or cubic polynomial. The general form of a cubic polynomial is ax3 + bx2 + cx + d, where a ≠ 0. Maximum number of terms of a cubic polynomial is four.

#### Zero / Root of a Polynomial

A number 'a' is said to be a zero of a polynomial p(x), if p(a) = 0. It is a solution to the polynomial equation p(x) = 0. If we draw the graph of p(x) = 0, the values where the curve cuts the x-axis are called the zeros of the polynomial.
A non-zero constant polynomial has no zero. Every real number is a zero of the zero polynomial.

Zero of a polynomial can be found by 'trial and error' method or by 'equating the polynomial to zero' method. In 'equating polynomial to zero' method, the zero of the polynomial can be found by making 'x' as the subject.

### Remainder Theorem

Let p(x) be a polynomial in x of degree greater than or equal to 1 and 'a' be any real number. If p(x) is divided by (x – a), then the remainder is p(a).

If p(x) is divided by (x – a), then the remainder is r(x) and the quotient is q(x). Thus, p(x) = (x - a) q(x) + r(x).

The degree of r(x) is always less than the degree of (x - a). Since the degree of (x - a) is one, the degree of r(x) is zero i.e. r(x) is a constant.

So, for every value of x, r(x) = r. Therefore, p(x) = (x – a) q(x) + r

Based on the remainder theorem:
• If a polynomial p(x) is divided by (x + a), then the remainder is p(– a).
• If a polynomial p(x) is divided by (ax – b), then the remainder is p(ba).
• If a polynomial p(x) is divided by (ax + b), then the remainder is p(-ba).
• If a polynomial p(x) is divided by (b – ax), then the remainder is p(ba).

### Factor Theorem

Let p(x) be a polynomial of degree n>1 and 'a' be a real number. If p(a) = 0, then (x – a) is a factor of p(x). Conversely p(a) = 0, if (x – a) is a factor of p(x).

If p(x) is a polynomial of degree n>1, then p(x) = (x - a) . q(x) + p(a).

If p(a) = 0, then p(x) = (x – a) . q(x). This proves that (x – a) is a factor of p(x).
• (x + a) is a factor of a polynomial p(x), if and only if p(– a) = 0
• (ax – b) is a factor of a polynomial p(x), if and only if p(b/a) = 0
• (ax + b) is a factor of a polynomial p(x), if and only if p(– b/a) = 0
• (x – a)(x – b) is a factor of a polynomial p(x), if and only if p(a) = 0 and p(b) = 0.

### Factorization of Polynomials Using Algebraic Identities

If g(x) and h(x) are two polynomials whose product is p(x). This can be written as p(x) = g(x) . h(x). g(x) and h(x) are called the factors of the polynomial p(x).

The process of resolving a given polynomial into factors is called factorisation. A non-zero constant is a factor of every polynomial.

### Algebraic Identities

Polynomials can be factorised using algebraic identities.
A polynomial of degree two is called a quadratic polynomial. The identities used to factorise the quadratic polynomials are:
• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2
• a2 – b2 = (a + b)(a – b)
• (x + a)(x + b) = x2 + (a + b)x + ab
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

A polynomial of degree three is called a cubic polynomial. The algebraic identities used in factorising a cubic polynomial are:
• (a + b)3 = a3 + b3 + 3ab (a + b)
• (a – b)3 = a3 – b3 – 3ab (a – b)
• a3 + b3 = (a + b)(a2 – ab + b2)
• a3 – b3 = (a – b)(a2 + ab + b2)
• a3 + b3 + c– 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)