'''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
- Interpreting What are Zeros of a polynomial function?
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
- Identifying the degree of a Polynomial function
- Evey polynomial function is interpreted algebraically,numerically and Graphically
- Every polynomial function generates different values for different values of x
- Every polynomial function can be plotted on a graph and the behaviour of the graph varies with the polynomial
- If the product of two linear expression is a polynomial,then that polynomial function can be factorised as (x+a)(x+b)=+x(a+b)+ab
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
- Factorising by inspection is based on the backwards or indirect use of the identity
- Factorising by spliting the middle term and grouping
- Equating Polynomial to zero
- Finding the values of x
- 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
- Understanding that given expression is a polynomial function
- Can all expressions be polynomial function?(eg. 1/x+1)
- what is an equation?what is a polynomial equation?(1/x+1=2 Is it a polynomial equation?)
- Understanding that a polynomial is a Function Click to know more about function
- Denoting a polynomial function as p(x) or f(x)
- Factorising a polynomial function
- Zero product principle
- Basic operations
For graphical interpretation,student should know
- 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
- 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
- Why should we equate a polynomial function to zero?
- What happens if we equate to any other number other than zero?
- Can we find Fives ,Eights ...of a polynomial function
- Can I find zeros of a polynomial function without factorising?
- Can all the polynomials be factorised to linear factors?
- when do we get linear and non linear graph
- Is A= is a polynomial function where A is the Area and s is the side of Square .Can we plot this on a graph?
- Plot p=4s where p is the perimeter and s is the side length of a square.Find the Zeroes of both the graphs