# '''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)=${\displaystyle x^{2}}$+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)=${\displaystyle x^{2}}$+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

## Contents

### 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=${\displaystyle s^{2}}$  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