Introduction (From Past)

Variable:

• Something that has ability to change is called variable.
• The value of variable is not fixed.
• We use alphabets  to show variables like x, y z, a, b, c, l, m etc.
• A variable allows us to express relation in any practical situation and to express many common rules and properties of geometry, algebra etc

Constant:

• Something which has fix value is called as constant e.g. 2, 5, 10, π, etc.

Term:

In an expression (e.g \(\displaystyle 5{{x}^{2}}-3x+5\)). It has  parts which are formed separately and then added. Such parts of an expression(e.g. \(\displaystyle 5{{x}^{2}}; -3x;  5\)) which are formed separately first and then added are known as terms.

Note: A term has four part. (i) sign (ii) numeral (iii)  variable (iv) power of variable.

Coefficients:

• The numerical factor is said to be the numerical coefficient or simply the coefficient of the term.

Example: 

• In 7xy, 7 is the coefficient of the term. It is also the coefficient of xy.
• In the term \(\displaystyle \frac{3}{5}xyz\), \(\displaystyle \frac{3}{5}\) is the coefficient of xyz,
• Similarly in the term \(\displaystyle -2\sqrt{3}{{x}^{3}}{{y}^{2}}\),\(\displaystyle -2\sqrt{3}\) is the  coefficient of \(\displaystyle {{x}^{3}}{{y}^{2}}\)

Algebraic expression and Polynomials

Expressions, containing arithmetical numbers, variables and symbols of operations are called algebraic expressions and polynomial are special case of algebraic expression. We will start off with polynomials in one variable.

Polynomials

Polynomials in one variable are algebraic expressions that consist of terms in the form of

\(\displaystyle \begin{array}{l}p\left( x \right)={{a}_{n}}{{x}^{n}}+{{a}_{{n-1}}}{{x}^{{n-1}}}+{{a}_{{n-2}}}{{x}^{{n-2}}}+……+{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+{{a}_{0}}\\where\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n\to \text{a}\,\text{non-negative}\,\text{integer(whole}\,\text{number}\text{)}\\\text{ }{{\text{a}}_{n}},{{\text{a}}_{{n-1,\,…..}}},\,{{\text{a}}_{2}},\,{{\text{a}}_{1}},{{\text{a}}_{0}}\,\to \,\text{Real numbers and are called as coefficient of terms}\end{array}\) 

\(\displaystyle {{a}_{n}}{{x}^{n}}\to \,The\,\,Leading\,Terms\,\,\And \,\,{{a}_{n}}\to \,The\,leading\,Coefficient\)

Example: \(\displaystyle p\left( x \right)=3{{x}^{4}}+5{{x}^{3}}-{{x}^{2}}+7x+10\)

here, \(\displaystyle n=4\,\,\,\And \,\,\,{{a}_{4}}=3,\,\,{{a}_{3}}=5,\,{{a}_{2}}=-1,{{a}_{1}}=7,\,{{a}_{0}}=10\)

In other word polynomial can also define as

“An algebraic expression in which involved variables,  have only non-negative integral (whole number) power is called a polynomial.

Degree

The degree of a polynomial in one variable is the largest exponent in the polynomial.

 Examples:\(\displaystyle p\left( x \right)=5{{x}^{3}}-{{x}^{2}}+7{{x}^{5}}+10\)      degree = 5

  \(\displaystyle p\left( x \right)=\sqrt{2}x+7{{x}^{2}}+10\) ,  degree = 3

Note:

1. We will often drop the “in one variable” part and just say polynomial.
2. It is denoted by p(x) or f(x) or g(x) or q(x) or r(y) etc.
3. A polynomial doesn’t have to contain all powers of x. e.g. \(\displaystyle p\left( x \right)=8{{x}^{3}}-27\),   here terms containing \(\displaystyle {{x}^{2}}\,\And \,x\) are missing.
4. Algebraic expressions in which the power of variable  are not whole number (i.e 0,1,2,3….) are not polynomials

Example: \(\displaystyle p\left( x \right)={{x}^{{\text{-3}}}}-7{{x}^{2}}+3x+10\,\) {Power of variable is negative}

\(\displaystyle p\left( x \right)=\sqrt{{2x}}+7{{x}^{2}}+10={{\left( {2x} \right)}^{{\frac{\text{1}}{\text{2}}}}}+7{{x}^{2}}+10\) {Power of variable is fraction (surd)}.

All the exponents in the algebraic expression must be non negative integers in order for the algebraic expression to be a polynomial.

5.   This doesn’t mean that radicals and fractions aren’t allowed in polynomials. They just can’t involve in the variables.

      For example, \(\displaystyle p\left( x \right)=\sqrt[3]{3}{{x}^{4}}+\frac{5}{7}{{x}^{3}}-8{{x}^{2}}+\frac{{10}}{{\sqrt{3}}}\)

There are lots of radicals and fractions in this algebraic expression, but the denominators of the fractions are only numbers and the radicands of each radical are only a number.  Each x in the algebraic expression appears in the numerator and the exponent is a non negative integer. Therefore, it is a polynomial.

Type of Polynomials

Polynomials can be divided on two basis-

1. 1. On the basis of Terms
   2. On the basis of Degree

On the basis of Terms

(a) Monomials: An expression with only one term is called a monomial.

Example: \(\displaystyle \frac{3}{4}x,\,\,\,\sqrt{5}{{x}^{2}},\,\,-2\sqrt{3},\,5{{y}^{3}}\) etc.

(b) Binomials: An expression with only two term is called a monomial.

      Example: \(\displaystyle \frac{3}{5}x+5,\,\,\,\,5{{y}^{3}}+\sqrt{2}y,\,\,\,\,5{{a}^{4}}-15{{a}^{2}}\,\) etc.

(c) Trinomials: An expression which contains three terms is called a trinomial.

      Example: \(\displaystyle 2{{x}^{2}}+5x+7,\,27{{x}^{3}}+15{{x}^{2}}-12\) etc

On the basis of Degree

(a) Constant Polynomials:

• A polynomial of degree zero is called as constant  polynomial.
• Standard form: \(\displaystyle P\left( x \right)=c\,;\,\,c\to \) Real Number, \(\displaystyle \,c\ne 0.\)

Example: p(x) = -5,   p(x) = 3/4    p(x) = π etc.

(b) Linear Polynomials: 

• A polynomial of degree one is called as constant  polynomial.
• Standard form: \(\displaystyle P\left( x \right)=ax+b\,;\,where\,\,a\ne 0;\,\,a,\,b\to \text{Real}\,\text{number}\text{.}\)

\(\displaystyle \begin{array}{l}\text{a}\to \text{Coefficient of x}\\b\to \text{Coefficient of }{{\text{x}}^{0}}\,\text{or}\,\text{constant term}\end{array}\)

Example: \(\displaystyle 3x+5,\frac{3}{7}x-\frac{1}{2},\,\,\sqrt{5}x\) etc.

(c) Quadratic Polynomials: 

• A polynomial of degree two is called as constant  polynomial.
• Standard form: \(\displaystyle P\left( x \right)=a{{x}^{2}}+bx+c\,;\,where\,\,a\ne 0;\,\,a,\,b,c\to \text{Real}\,\text{number}\text{.}\)

\(\displaystyle \begin{array}{l}\text{a}\to \text{Coefficient of }{{\text{x}}^{2}}\\b\to \text{Coefficient of x}\,\\c\to C\text{onstant term}\end{array}\)

Example: \(\displaystyle 2{{x}^{2}}+3x+5,\,\,\,\sqrt{2}{{x}^{2}}-3\sqrt{2}x,\,\,\,\,4{{x}^{2}}-25\) etc.

(d) Cubic Polynomials: 

• A polynomial of degree three is called as constant  polynomial.
• Standard form: \(\displaystyle P\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d\,;\,where\,\,a\ne 0;\,\,a,\,b,c,d\to \text{Real}\,\text{number}\text{.}\)

\(\displaystyle \begin{array}{l}\text{a}\to \text{Coefficient of }{{\text{x}}^{3}}\\b\to \text{Coefficient of }{{\text{x}}^{2}}\,\\c\to \text{Coefficient of x}\,\\d\to \text{Constant term}\end{array}\)

Example: \(\displaystyle {{x}^{3}}-1,\,\,3{{x}^{3}}+4{{x}^{2}}+5x+10,\,\,{{x}^{3}}-\sqrt{2}{{x}^{2}}+15\)

(e) Biquadratic Polynomials: 

• A polynomial of degree four one is called as constant  polynomial.
• Standard form:\(\displaystyle P\left( x \right)=a{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+e\,;\,where\,\,a\ne 0;\,\,a,\,b,c,d,e\to \text{Real}\,\text{number}\text{.}\)

\(\displaystyle \begin{array}{l}\text{a}\to \text{Coefficient of}\,{{\text{x}}^{4}}\\b\to \text{Coefficient of }{{\text{x}}^{3}}\\c\to \text{Coefficient of }{{\text{x}}^{2}}\,\\d\to \text{Coefficient of x}\,\\e\to \text{Constant term}\end{array}\)

Example:\(\displaystyle 4{{x}^{4}}-5x+6,\,\,{{x}^{4}}+7{{x}^{3}}-3{{x}^{2}}+10x-12,\,\,{{x}^{4}}-5\sqrt{2}{{x}^{2}}+15\)

Note: \(\displaystyle \mathbf{p}\left( \mathbf{x} \right)=\mathbf{0}\) is called as Zero  polynomial.  Its degree is undefine.

\(\displaystyle \text{p}\left( \text{x} \right)\text{=0=0 }\!\!\times\!\!\text{ }{{\text{x}}^{\text{2}}}\text{=0 }\!\!\times\!\!\text{ }{{\text{x}}^{\text{5}}}\text{=0 }\!\!\times\!\!\text{ }{{\text{x}}^{{\text{100}}}}\)

Here, we cannot decide the degree,  that is why degree of zero polynomial is not define.

Zeroes of Polynomials

• The value(s) of variable (x) at which a polynomial become zero is(are)  called as zeroes of polynomial.
• A real number ‘x= k’ is a zero of a polynomial p(x), if p (k) = 0.
• A non-zero constant polynomial has no zero.
• Every real number is a zero of the zero polynomial {p(x)=0}.

Example: Check whether –2 and 2 are zeroes of the polynomial \(\displaystyle {{x}^{2}}-3x-10\).

Sol: Let p(x) = \(\displaystyle {{x}^{2}}-3x-10\).

Case (i) x = 2

Then, \(\displaystyle \begin{array}{l}p\left( 2 \right)={{2}^{2}}-3\left( 2 \right)-10\\=\,4-6-10\\=4-16\\p\left( 2 \right)=-12\end{array}\)

since, \(\,p\left( 2 \right)\ne 0\)

Therefore, 2 is not a zero of the polynomial \(\displaystyle {{x}^{2}}-3x-10\).,

Case (ii) x = -2

\(\begin{array}{l}p\left( {-2} \right)={{\left( {-2} \right)}^{2}}-3\left( {-2} \right)-10\\=\,4+6-10\\=10-16\\p\left( 2 \right)=0\end{array}\)

since, \(\,p\left( 2 \right)=0\)

Therefore, -2 is a zero of the polynomial \(\displaystyle {{x}^{2}}-3x-10\).,