Polynomials 

• Polynomials:-

          Polynomials are the expressions which have variables and their coefficients ,and powers of variables must be always whole number.

Ex:- p(x)=x²+3x+4

Note:- powers of all variables of polynomials must be 0 or positive integers.

Example:- 3x‐²+2x+3 is not a polynomial 

Degree of polynomials 

The highest power of the polynomial is called the Degree of polynomial. 

Example:- p(x)=4x³+3x²+5x+1

                   Here the highest power of x is 3 so the Degree of polynomial is 3.

                   P(x)=100x+10

                   Here the highest power of x is 1 so the Degree of polynomial is 1

Note:-Degree of constant is 0

                   P(x)=100

Types of polynomials 

1. Linear polynomial 

2.quadratic polynomial 

3.cubic polynomial 

1. Linear polynomial:-

     If the Degree of the polynomial is 1 called linear polynomial. 

Example:- p(x):-3x+4

The general form of a linear polynomial is 

p(x)=ax+b

• Linear will have only one solution 
• Linear equation will give straight line graph 
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2.quadratic polynomial 

      If the Degree of polynomial is 2 then it called quadratic polynomial. 

Example:-p(x) =x²+3x+4

• The quadratic polynomials have 2 solutions ,As it has Degree 2.
• The general form of quadratic polynomial is p(x)=ax²+bx+c
• The graph of quadratic equation is parabola.
• Test yourself
  

3. Cubic polynomial 

     If the Degree of polynomial is 3 then it called Cubic polynomial.

Example:- p(x)=5x³+2x+3

• The cubic polynomial has 3 solutions 
• The general form of cubic polynomial p(x)=ax³+bx²+cx+d

Points to find given expressions are polynomial or not

• Power of given expressions always should positive integers or zero.
• Example:-3x‐²+4x+3 is not a polynomial because it has negative power
• Power of variables of given expressions must be rational.
• Example:-`p(x)=5x^\sqrt3+3x+5`here the power of x is irrational number Therefore it is not a polynomial 
• Variables should not be in denominator 
• Example`p(x)=\frac2x+3x+1`here the Variable x is in denominator there it is not a polynomial 
• Variable should not be irrational number
• Example:-`p(x)=\sqrt x+2x+1`here the Variable is under root it is irrational, it is not a polynomial 

Finding value of polynomials 

1. Find value of polynomial `p(x)=x²+3x+2` If x=2

Sol. `p(x)=x^2+3x+2`

         `p(x)=2^2+3(2)+2`

           `p(x)=4+6+2`

              `p(x)=12`

2. Find value of polynomial p(x)=3x²+9x+6 if it's zeroes are 1 and 2

Sol. If x=1

        p(x)=3x²+9x+6

        P(1)=3(1)²+9(1)+6

        P(1)=3+9+6

        P(1)=18

Test yourself

Finding zero of polynomial 

1. Find zero of polynomial p(x)=3x+1.

Sol. p(x)=3x+1

         0 = 3x+1

         3x=-1

          x=`-frac(1)2`

2. Find zero of polynomial p(x)=5x

Sol. p(x)=5x

        0=5x

        x =`frac(0)5`

        x=0

3. Find zeroes of polynomial p(x)=x²+4x+4

Sol. x²+4x+4=0

       x²+2x+2x+4=0 

       x(x+2)+2(x+2)=0

       (x+2)(x+2)=0

        x+2=0. Or x+2=0

        x=-2.   Or  x=-2

4. Find zeroes of p(x]=x²+7x+10

Sol. x²+7x+10=0

       x²+5x+2x+10=0

       x(x+5)+2(x+5)=0

       (x+5)(x+2)=0

         x+5=0. Or.  x+2=0

         x=-5.    Or.   x=-2

To be continued: