'''Solution for the Problems of Chapter 8-Polynomials 10 STD'''

Find the zeroes of the following polynomials and verify the result.
We can solve the exercise problems graphically visualising

Interpretation of the given problem

1. Interpreting What are Zeros of a polynomial function?

Algebraic interpretation
What do we mean by a root, or zero, of a polynomial function? It is a solution to the polynomial equation, P(x) = 0. It is that value of x that makes the polynomial function equal to 0

1. Identifying the degree of a Polynomial function
2. Evey polynomial function is interpreted algebraically,numerically and Graphically
3. Every polynomial function generates different values for different values of x
4. Every polynomial function can be plotted on a graph and the behaviour of the graph varies with the polynomial
5. If the product of two linear expression is a polynomial,then that polynomial function can be factorised as (x+a)(x+b)=$x^{2}$+x(a+b)+ab

Graphical Interpretation
Zeros of a polynomial or Roots of a polynomial function is that value of x where the graph crosses or touches the x-axis. ( At the x-intercept on the x-axis y = 0)

Methods/Steps for Solving

Algebraic method:

• Factorising by inspection is based on the backwards or indirect use of the identity

(x+a)(x+b)=$x^{2}$+x(a+b)+ab

• Factorising by spliting the middle term and grouping
• Equating Polynomial to zero
• Finding the values of x

Graphical method

• every function is a relationship of x and p(x) values, we can create a table of values for any polynomial function, these are just the values that can be plotted on a coordinate axes. In other words, a table of values is simply some points with (x,p(x)) as coordinates.Find the value of x for which p(x) becomes zero

Learner's previous knowledge

For algebraic interpretation,Students should know

1. Understanding that given expression is a polynomial function
2. Can all expressions be polynomial function?(eg. 1/x+1)
3. what is an equation?what is a polynomial equation?(1/x+1=2 Is it a polynomial equation?)
4. Understanding that a polynomial is a Function Click to know more about function
5. Denoting a polynomial function as p(x) or f(x)
6. Factorising a polynomial function
7. Zero product principle
8. Basic operations

For graphical interpretation,student should know

1. about plotting the points on a graph sheet

Concepts to be built

• A polynomial function can have one Zero or two or multiple Zeros.
• Alinear polynomial has single Zero,a Quadratic polynomial has Double Zeros and so on
• This can be found by number of values we get in factorisation or by investigating how many times a graph touces x-axis(if x is a real vale?)
• Record the Observations

Can we look at the table values to analyse a graph? Analysing Quadratic polynomial function based on its symmetry

Skill to be built

1. skill of Factorisation
   

Identifying gaps to be filled

• identifying degree of a polynomial function
• Basic mathematical operation concept for factorising or for tabulating the vales of x and p(x)
• difficulty in corelating algebraic and graphical and numeric interpretation

Provide an algorithm

Use the Geogebra applet given below to visualise the Zeros of the problems of Chapter 8

For Thought provoking

1. Why should we equate a polynomial function to zero?
2. What happens if we equate to any other number other than zero?
3. Can we find Fives ,Eights ...of a polynomial function
4. Can I find zeros of a polynomial function without factorising?
5. Can all the polynomials be factorised to linear factors?
6. when do we get linear and non linear graph
7. Is A=$s^{2}$ is a polynomial function where A is the Area and s is the side of Square .Can we plot this on a graph?
8. Plot p=4s where p is the perimeter and s is the side length of a square.Find the Zeroes of both the graphs