REAL NUMBERS,POLYNOMIALS,TRIGNOMETRY etc. For class 10th

1. Given positive integers
a and b, there exist
unique integers q
and r satisfying a =
bq + r, 0 ≤ r < b.

2.  RELATIONSHIP BETWEEN ZEROS AND
COEFFICIENT OF A POLYNOMIAL

relationship between zeros and coefficient of a polynomial in case of
quadratic and cubic polynomial is stated as follows

(1) QUADRATIC POLNOMIAL

Let ax² +bx +c be the quadratic polynomial and α and β are its zeros ,then

Sum of zeros = α + β = -b/a = - (coefficient of x)/ (coefficient of x²)

Product of zeros = αβ = c/a = constant term / (coefficient of x²)

If we need to form an equation of degree two ,when sum and products of
the roots is given ,then K[x²-( α + β )x + αβ ]=0 is the required equation
,where k is constant .

3. Procedure for finding zeros of
a quadratic polynomial
· Find the factors of the quadratic
polynomial .
· Equate each of the above factors
(step 1) with zero.
· Solve the above equation (step 2)
· The value of the variables obtained
(step 3) are the required zeros .

4. (2) CUBIC POLYNOMIAL
Let axᶟ +bx² +cx +d be the cubic
polynomial and α , β and γ are its zeros
,then
Sum of zeros = α + β + γ = -b/a = -
(coefficient of x²)/ (coefficient of xᶟ)
Sum of Product of zeros taken two at a
time = αβ +βγ +γα = c/a = (coefficient of
x)/ (coefficient of xᶟ)
Product of zeros = αβγ = -d/a = - constant
term / (coefficient of xᶟ)
When sum of zeros , Sum of Product of
zeros taken two at a time , Product of zeros
is given , then K[xᶟ-( α + β + γ )x² + (αβ +βγ
+γα)x – αβγ ]=0 is the required equation
,where k is constant,

5. Procedure for finding
zeros of a cubic
polynomial
· By hit and trial method find one
zeros of the polynomial using remainder
theorem
· Now if we know one zero , then we
know one factor of the polynomial . divide
the cubic polynomial by this factor to
obtain quadratic polynomial
· Now , solve this quadratic
polynomial to obtain the other two zeros of
the cubic polynomial .
· These three zeros are the required

6. One algebraic method is the substitution
method. In this case, the value of one
variable is expressed in terms of another
variable and then substituted in the
equation. In the other algebraic method –
the elimination method – the equation is
solved in terms of one unknown variable
after the other variable has been eliminated
by adding or subtracting the equations. For
example, to solve:
8x + 6y = 16
-8x – 4y = -8

7. Using the elimination method, one would
add the two equations as follows:
8x + 6y = 16
-8x – 4y = -8
2y = 8
y = 4
The variable "x" has been eliminated. Once
the value for y is known, it is possible to
solve for x by substituting the value for y
in either equation:
8x + 6y = 16
8x + 6(4) = 16
8x + 24 = 16
8x + 24 – 24 = 16 – 24
8x = -8
X = - 1

9.  Basic Proportionality Theorem states that if a
line is drawn parallel to one side of a triangle
to intersect the other 2 points , the other 2
sides are divided in the same ratio.
 It was discovered by Thales , so also known
as Thales theorem.
Basic Proportionality
Theorem

10. ProvingtheThales’
Theorem

11. If a line divides any two sides of a
triangle in the same ratio, then the line is
parallel to the third side

12. Provingtheconverseof
Thales’Theorem

13. Trigonometry means
“Triangle” and “Measurement”
Introduction Trigonometric Ratios

14. There are 3 kinds of trigonometric
ratios we will learn.
sine ratio
cosine ratio
tangent ratio

15.  Definition of Sine Ratio.
 Application of Sine Ratio.

16. Definition of Sine Ratio.

1
If the hypotenuse equals to 1
Sin = Opposite sides

17. Definition of Sine Ratio.

For any right-angled triangle
Sin =
Opposite side
hypotenuses

18.  Definition of Cosine.
 Relation of Cosine to the sides of right angle
triangle.

19. Definition of Cosine Ratio.

1
If the hypotenuse equals to 1
Cos = Adjacent Side

20. Definition of Cosine Ratio.

For any right-angled triangle
Cos =
hypotenuses
Adjacent Side

21.  Definition of Tangent.
 Relation of Tangent to the sides of right
angle triangle.

23. Definition of Tangent Ratio.

For any right-angled triangle
tan =
Adjacent Side
Opposite Side

24. hypotenuse
sideopposite
sin 
hypotenuse
sidedjacenta
cos 
sidedjacenta
sideopposite
tan 
Make Sure
that the
triangle is
right-angled