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Looking at Indices

Extracts from this document...

Introduction

Maths AS GURU

P1 ALGEBRA

Looking at Indices

The manner in which we count is based on the number of fingers (digits) that we have.

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image01.png

Our number system is the product of centuries of development . The symbols originated with the Hindus and the name from the Romans. Decimal, means tenth or tithe.

image00.png

image42.png

The numbers are put together so that the posi ...read more.

Middle



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image69.png

The algorithm was presented without proof to avoid others claiming the idea as their own. The method is clearly the same as the formula method of solution.
He showed the solution of particular equations, for example:
x2 - 10x = -9.
He called the numbers by the names:

  • 1 is "the square"
  • -10 is "the unknown"
  • -9 is "the absolute number"



image00.png

image70.png

Top

Brahmagupta's work yields only one solution for each quadratic equation. However, by the 12th century, scholars were identifying both solutions.
Problems were now being written to teach mental skills rather than to solve problems from astronomy.

image00.png

Solve:
x2 - 10 = -9
(-9) × 4 + (-10) × (-10)= 64
root 64 = 8
8 - (-10) = 18
18 ÷ 2 × 1
x = 9

Another Indian scholar, Bhaskara set a problem such as this one:

The fifth part less three, squared, of a troupe of monkeys had gone to a cave and one monkey was in sight having climbed a tree. Say how many monkeys there are.

Top

Although the two values of the unknown were described by Bhaskara, one value was dismissed as incongruous.

If the number of monkeys is 5, the number going to the cave is negative. This is clearly not a solution to the riddle. What is the only

reasonable solution?

The monkey problem can be written as a modern equation:

image71.pngwhich, rewritten, becomes x2 - 55x + 250 = 0

These ideas were translated into symbols in the 17th Century and should be familiar to all students of mathematics.

image00.png

ax2 + bx + c = 0

x =

-b ± image35.png(b2 - 4ac)

image21.png

2a

Factorisation

If the product of two numbers p and q equals zero, then either p or q (or both) are zero.

image00.png

If

p × q

= 0

then

either

p

= 0

or q

= 0

orp

= q = 0

This means that if a quadratic equation can be factorised, it is possible to extract linear equations which lead to solutions. These solutions are often referred to as roots of the equation.

...read more.

Conclusion

x - 2.3) = x2- 3.8x + 3.45
the quadratic function can be form

Quadratic Functions

This is a quadratic function.

When
x is 3 this quadratic function is zero.

image00.png

f(x)

=

x2 - 4x + 3

x = 3

f(3)

=

32 - 4 × 3 + 3

=

9 - 12 + 3

=

0

The Factor Theorem states that if a function of xis zero when x = 3, then (x - 3) is a factor of the function.

image00.png

f(3) = 0
image18.png
(x - 3)
is a factor of
f(x) = x2 - 4x + 3

The other factor can then be found to give the completely factorised form of the quadratic.

image00.png

By multiplication it is found that the other factor is (x -1) and
f(x) = (x - 3)(x -1)

Top

The Factor Theorem states that if f(a) = 0 then (x - a) is a factor of the function f(x).

image00.png

(x - a) is a factor of f(x) because
f(a) = 0

The process can be used in reverse to find a quadratic function with particular properties.
Find the function with factors
(x - 3) and (x + 5) 

image00.png

The function with factors (x -3) and (x + 5) is

f(x)

=

(x - 3)(x + 5)

=

x2 + 2x - 15

Sometimes functions with higher powers of x can be factorised by this method .

The process of obtaining the factors becomes longer and sometimes involves long-division.

f(x) = x3 - 12x2 + 47x - 60
f(5) = 0
(x - 5)
is a factor.

Top

Long Division

Long division takes practise. Here is an example:

image00.png

Divide:
f(x) = x3 - 12x2 + 47x - 60 by the factor (x -5) to find the other factors

image34.png

x cubed divided by x gives x squared
multiply the factor
(x -5) by x2 to give x3 - 5x2
subtract this from the first two terms and then divide the answer by
x
This gives
-7x and after multiplication 7x2 + 47x
Subtract to leave
12x which when divided by x gives 12

Long division is not essential. Other factors can be found using the Factor Theorem again.
Factorising continues with the quadratic factor.

Factorise the cubic function

f(x)

=

x3 - 12x3 + 47x - 60

=

(x -5)(x2 - 7x + 12)

=

(x -5)(x - 3)(x - 4)

...read more.

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