# Polynomials

## Polynomials Synopsis

**Synopsis**

- A
**polynomial**p(x) in one variable x is an algebraic expression in x of the form

p(x)= a_{n}X^{n}+a_{n-1}X^{n-1}+a_{n-2}X^{n-2}+……+a_{2}X^{2}+a1X + a0 , where

- a
_{0},a_{1},a_{2}….an are constants - x is a variable
- a
_{0},a_{1},a_{2}….an are respectively the coefficients of xi - Each of a
_{n}X^{n}+a_{n-1}X^{n-1}+a_{n-2}X^{n-2}+……+a_{2}X^{2}+a1X + a0 with an ≠0 ,is called a term of a polynomial.

**Degree of a polynomial**

The highest power of the variable in a polynomial is called the**degree**of the polynomial.

In case of more than one variable, degree is the highest sum of the powers of the variables.- A polynomial with one term is called a
**monomial.** - A polynomial with two terms is called a
**binomial.** - A polynomial with three terms is called a
**trinomial.** - A polynomial with degree zero is called a
**constant polynomial**. For example: 1, -3. The degree of non-zero constant polynomial is zero - A polynomial of degree one is called a
**linear polynomial**. It is of the form ax + b. For example: x - 2, 4y + 89, 3x - z. - A polynomial of degree two is called a
**quadratic polynomial**. It is of the form ax2 + bx + c. where a, b, c are real numbers and a ¹ 0 For example: x2 - 2x+ 5 etc. - A polynomial of degree three is called a
**cubic polynomial**and has the general form ax3 + bx2 + cx +d. For example: x3 - 2x2 -2x+5 etc. - A
**bi-quadratic polynomial**p(x) is a polynomial of degree four which can be reduced to quadratic polynomial in the variable z = x2 by substitution. - The constant polynomial 0 is called the
**zero polynomial**. Degree of zero polynomial is not defined. - The value of a polynomial f(x) at x = p is obtained by substituting x = p in the given polynomial and is denoted by f(p).
- A real number ‘a’ is a zero or root of a polynomial p(x) if p (a) = 0.
- The number of real zeroes of a polynomial is less than or equal to the degree of polynomial.
- Finding a zero or root of a polynomial f(x) means solving the polynomial equation f(x) = 0.
- A non-zero constant polynomial has no zero.
- Every real number is a zero of a zero polynomial.

**Division of a polynomial by another polynomial**

If p(x) and g(x) are the two polynomials such that degree of p(x) ³ degree of g(x) and g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that:

p (x) = g(x) q(x) + r(x)

Where, r(x) =0 or degree of r(x) < degree of g(x).

**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 remainder is p(a).

- If the polynomial p(x) is divided by (x + a), the remainder is given by the value of p (-a).

**Factor theorem**

Let p(x) is a polynomial of degree n ≥ 1 and a is any real number such that p(a) = 0, then (x – a) is a factor of p(x).

**Converse of factor theorem**

Let p(x) is a polynomial of degree n ≥ 1 and a is any real number. If (x – a) is a factor of p(x), then p(a) = 0.

- (x + a) is a factor of a polynomial p(x) if p(–a) = 0.
- (ax – b) is a factor of a polynomial p(x) if p(b/a) = 0.
- (ax + b) is a factor of a polynomial p(x) if p(–b/a) = 0.
- (x – a)(x – b) is a factor of a polynomial p(x) if p(a) = 0 and p(b) = 0.

- For applying factor theorem, the divisor should be either a linear polynomial of the form (ax + b) or it should be reducible to a linear polynomial.

- A quadratic polynomial ax
^{2}+ bx+ c is**factorised by splitting the middle term**by writing b as ps + qr such that (ps) (qr) = ac.

Then, ax^{2}+ bx+ c = (px + q) (rx + s)

- An
**algebraic identity**is an algebraic equation which is true for all values of the variables occurring in it.

- Some useful
**quadratic identities**:

- (x+y)
^{2}= X^{2}+ 2xy + y^{2} - (x-y)
^{2}= X^{2}- 2xy + y^{2} - (x-y)(x+y)=x
^{2}-y^{2} - (x-a)(x+b)=x
^{2}+(a+b)x+ab - (x+y+z)
^{2}= x^{2}+y^{2}+z^{2}+2xy+2yz+2zx

Here x, y, z are variables and a, b are constants.

- Some useful
**cubic identities:**

- (x+y)
^{3}= x^{3}+ y^{3}+ 3xy(x+y) - (x-y)
^{3}= x^{3}- y3- 3xy(x-y) - x3 +y
^{3}=(x + y)(x^{2}- xy + y^{2}) - x3 - y
^{3}= x - y)(x^{2}+ xy + y^{2}) - x3 +y
^{3}+z^{3}– 3xyz + (x+y+z) (x^{2}+y^{2}+z^{2}– xy- yz –zx) - if x+y+z + 0 then x
^{3}+ y^{3}+z^{3}+ 3xyz

Here, x, y and z are variables.

**Synthetic Division for Factorisation**

The steps are as follows:

- Write the coefficients in the descending powers of x in the first horizontal row.
- The multiplier is obtained by putting the divisor x – a = 0
- Write down the leading coefficient in the bottom row just below it.
- Multiply ‘a’ with the value just written on the bottom row.
- Add the two values and write the sum in the bottom row.
- Repeat the steps (iii) and (iv) until the result turns out to be zero.

For example: Divide 3x^{3}- 2x^{2}- 19x +22 by (x-2)

Sol.

Therefore,

3x^{3}- 2x^{2}19x +22 =(x-2) (3x^{2}+4x-11)

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