#### Topics

##### Number Systems

##### Algebra

##### Geometry

##### Trigonometry

##### Statistics and Probability

##### Coordinate Geometry

##### Mensuration

##### Internal Assessment

##### Real Numbers

##### Pair of Linear Equations in Two Variables

- Linear Equation in Two Variables
- Graphical Method of Solution of a Pair of Linear Equations
- Substitution Method
- Elimination Method
- Cross - Multiplication Method
- Equations Reducible to a Pair of Linear Equations in Two Variables
- Consistency of Pair of Linear Equations
- Inconsistency of Pair of Linear Equations
- Algebraic Conditions for Number of Solutions
- Simple Situational Problems
- Pair of Linear Equations in Two Variables
- Relation Between Co-efficient

##### Arithmetic Progressions

##### Quadratic Equations

- Quadratic Equations
- Solutions of Quadratic Equations by Factorization
- Solutions of Quadratic Equations by Completing the Square
- Nature of Roots of a Quadratic Equation
- Relationship Between Discriminant and Nature of Roots
- Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
- Application of Quadratic Equation

##### Polynomials

##### Circles

- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Tangent to a Circle
- Number of Tangents from a Point on a Circle
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles

##### Triangles

- Similar Figures
- Similarity of Triangles
- Basic Proportionality Theorem (Thales Theorem)
- Criteria for Similarity of Triangles
- Areas of Similar Triangles
- Right-angled Triangles and Pythagoras Property
- Similarity of Triangles
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Triangles Examples and Solutions
- Angle Bisector
- Similarity of Triangles
- Ratio of Sides of Triangle

##### Constructions

##### Heights and Distances

##### Trigonometric Identities

##### Introduction to Trigonometry

##### Probability

##### Statistics

##### Lines (In Two-dimensions)

##### Areas Related to Circles

##### Surface Areas and Volumes

- Concept of Surface Area, Volume, and Capacity
- Surface Area of a Combination of Solids
- Volume of a Combination of Solids
- Conversion of Solid from One Shape to Another
- Frustum of a Cone
- Concept of Surface Area, Volume, and Capacity
- Surface Area and Volume of Different Combination of Solid Figures
- Surface Area and Volume of Three Dimensional Figures

#### notes

p(x)= value of x at which p(x)=0 is called zero of p(x) ,which geometrically means x cordinate of point where graph of y= p(x) cuts x axis. Here in this concept we are going to study how by using coefficeint we can tell the zeroes of polynnomial p(x).

1) Linear Polynomial- p(x)= ax+b, a is not equal to 0

Let ax+b= 0

ax=b

Hence x= `-b/a.` By substituing this in p(x) we get `p(-b/a)= a(-b/a)+b`

After observing carefully we can conclude that

zero of linear polynomial= -constant/ coefficient Example- Find the zero of `p(x)= 13x+9,` zero of `p(x)= -9/13`

Let's cross check is method, `p(-9/13)= 13(-9/13)+9= -9+9= 0`

Thus in case of Linear Polynomial `p(x)=ax+b`, where a is not equal to 0

Zero of p(x)= `-"constant"/ "coefficient"` also zero of `p(x)= "-b"/a`

2) Quadratic Polynomial- `p(x)= ax^2+bx+c`, here in quadratic polynomial the maximum numbers of zeros are two. So let's assume alpha and beta are two zeros of

`p(x)= ax^2+bx+c,`Then the `"Sum" "of" "zeros"= "-coefficient of x"/ "coefficient of x"^2` i.e `alpha+ beta= "-b"/a`

And `"Product" "of" "zeros"= "constant term"/ "coefficient of x"^2` i.e `alpha×beta= c/a`

Example 1- Find the zeros of the polynomial `p(x)= x^2+12x+35` and verfiy the relationship between the zeros and the coefficient.

Using Factorisation method to find factors of the given equation we get (x+7) and (x+5) as the factors of polynomial `x^2+12x+35`. If (x+7)= 0 then x= -7 also if (x+5)= 0 then x= -5, here if -7 and-5 are the zeros of the polynomial then let alpha be -7 and beta be -5

As we know `alpha×beta= "-coefficient of x"/"coefficient of x"^2` = `x^2=-12/1= -12`

and `alpha×beta= "constant term"/" coefficient of x"^2= 35/1= 35`

As assumed above alpha is -7 and beta is -5, thus `alpha+beta= -7+(-5)= -7-5= -12`

and `alpha xx beta= -7 xx -5= 35` Hence verified.

Example 2- Find a quadratic polynomial, the sum and product of which zeros are -3 and 2 respectively.

General quadratic polynomial is `p(x)= ax^2+bx+c`, where a is not equal to 0 and alpha and beta are the zeros of this polynomial.

As per the given condition, `alpha+beta= -3 and alpha×beta= 2`

`alpha+beta= "-b"/a`

`-3= "-b"/a`

`b=3a`

`alpha×beta= c/a`

`2= c/a`

`c= 2a`

by substitution we get, `p(x)= ax^2+bx+c= ax^2+ 3ax+ 2a`

taking a common `p(x)= a(x^2+3x+2)` Here, because a is not equal to zero, for different value of a we will have different value of polynomial. If take a=1 then `p(x)= x^2+3x+2`

Let's say for a=3, `p(x)= 3(x^2+3x+2)= 3x^2+9x+6`

`alpha+beta= "-b"/a= "-9"/3= -3`

`alpha×beta= c/a= 6/3= 2`

Hence verified with the given condition above.

So if we take a as any value we get `alpha+beta` and `alpha×beta` verified.

This example could also be solved using a formula i.e

p(x)= `x^2`- (sum of roots)x+ product of the roots

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

So we can say that if the sum and product of roots are given then by using p(x)= `x^2`- (sum of roots)x+ product of the roots, we could directly find the quadratic polynomial.

3) Cubic Polynomial- `p(x)= ax^3+bx^2+cx+d,` here in cubic polynomial the maximum numbers of zeros are three. So let's assume `alpha, beta and gamma` are the three zeros of `p(x)= ax^3+bx^2+cx+d`,

then `alpha+beta+gamma= "-b"/a,` `alpha×beta+ beta×gamma+ gamma×alpha= c/a`, and `alpha×beta×gamma= "-d"/a`

Let us try to understand how is it derived

In generic terms, `p(x)= ax^3+bx^2+cx+d`, where a is not equal to zero, and alpha, beta and gamma are the zero of the polynomial. If alpha, beta and gamma are the zeros of the polynomial then the roots of the polynomial will be `(x-alpha) (x-beta) and (x-gamma).`

Thus, `p(x)= k (x-alpha) (x-beta) (x-gamma)`, where k is any constant

`ax^3+bx^2+cx+d = k (x-alpha) (x-beta) (x-gamma)`

`ax^3+bx^2+cx+d =k [x^3- (alpha+beta+gamma)x^2+ (alpha×beta+ beta×gamma+gamma×alpha)x- alpha×beta×gamma]`

comparing the coefficient of LHS and RHS we get,

a=k

b= -k(`alpha+beta+gamma`) i.e sum of the roots

`alpha+beta+gamma = -b/k`

`alpha+beta+gamma`= `"-b"/a`

c= k(`alpha×beta+ beta×gamma+ gamma×alpha`) i.e product of zeros taken two at a time= `alpha×beta+ beta×gamma+ gamma×alpha`

= `c/k`

`alpha×beta+ beta×gamma+ gamma×alpha= c/a`

d= -k(`alpha×beta×gamma`) i.e product of zeros= `alpha×beta×gamma`= `-d/k`

`alpha×beta×gamma`= `"-d"/a`

This is the relationship betweem coefficient and zeros of cubic polynomials.

Example- `2x^3-5x^2-14x+8=0`

`alpha+beta+gamma= "-b"/a= "-(-5)"/2= 5/2`

`alpha×beta+ beta×gamma+ gamma×alpha= c/a= "-14"/2= -7`

`alpha×beta×gamma= "-d"/a= "-8"/2= -4`