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

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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

Geogebra Applet

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