Learning High School Algebra

Algebra is a branch of mathematics that deals with  the study of  rules of operations and relations. Diophantus is regarded as the father of algebra. Basic algebraic concepts include variables and constants, expressions, terms, polynomials, equations and algebraic structures. So much is known about the subject that study of this takes a life time. Algebra offers lot of food for thought for students and mathematics lovers will have lot of fun solving problems in algebra. The beauty of the subject is that it prompts the student to think indicatively to solve problems and helps to develop analytical and logical skills for the student. Study of mathematics at any level will cover some topics in algebra.

High school algebra covers topics such as polynomials, algebraic expressions and identities, equtaions and factorization.

Let us learn some solved problems in the above topics of high school algebra.


Learning Polynomials and Factorization in high school algebra:


At high school level, polynomials and factorization are important topics covered in algebra.

POLYNOMIALS:

An Algebraic expression of the form axn is called Monomial in x. For example, 7x3   .The sum of two monomials are called a Binomial and the sum of three monomials are called Trinomial. For example, 2x3 + 3x is a binomial and 2x5 – 3x2 + 3 is Trinomial. The sum of a finite number of monomials in x is called a polynomial in x.

Example:

Find the sum of 2x4 – 3x2 + 5x + 3 and 4x + 6x3 – 6x2 – 1.

Solution:

Using the associative and distributive properties of real numbers, we obtain

(2x4 – 3x2 + 5x + 3) + (6x3 – 6x2 + 4x – 1) = 2x4 + 6x3 – 3x2 – 6x2 + 5x + 4x + 3 – 1

= 2x4 + 6x3 – (3+6)x2 + (5+4)x + 2

= 2x4 + 6x3 – 9x2 + 9x + 2.  (answer)

FACTORIZATION :

The process of writing polynomial as a product of two or more simpler polynomials is called Factorization.

The way of writing a polynomial as a product of two or more simpler polynomials is called factorization. The process of factorization is also known as the resolution into factors.

Example 1: Factorize x2 – 2xy – x + 2y.

Solution:

x2 – 2xy – x + 2y = (x2 – 2xy) – (x – 2y)

= x(x – 2y) + (–1) (x – 2y)

= (x – 2y) [x + (–1)]

= (x – 2y) (x – 1).   (answer)


Example 2: Solve 9x2 = x.

Solution:

9x2 = x

9x2 − x = 0

x (9x − 1) = 0        ab = 0

x = 0      or    x =

What if we attempt the same problem using the another method?

9x2 = x

9x = 1 divide by x

x =

Every quadratic equations has 2 roots. Dividing the quadratic equation by x removes the root x = 0.

However, dividing by a constant does not impact the roots.

Learning Algebraic Identites in high school algebra:

Algebraic Identities:

Algebraic identities is important in High school algebra . Algebraic identities are algebraic equations satisfied by any value of the variables.
The algebraic identity of x + 0 = x tells us that anything (x) added to zero equals the original "anything," no matter what rate that "anything" (x) may be. Like normal algebra, Boolean algebra has its own unique identities base on the bivalent states of Boolean variables.

Example 4: Solve for variables x,y and z:

x + y + 2z = 2  ------> (1)

3x - y + 3z = 4  ------>(2)

2x + y + 4z = 6  ------>(3)

Solution:

Solve (1) and (2),

x + y + 2z = 2 ----> (1)

3x - y + 3z = 4 -----> (2)

add the above two equations.

we get 4x + 5z = 6    ------> (4)

solve (1) and (3)

(3) * 2 ---->        4x + 4y +8z = 12

(4)     ----->        4x + 0y +5z = 6
(-)           (-)      (-)

3z = 6

z = 2

substitute z = 2 in (4) eqn

4x + 5(2) = 6

4x = -4

x = -1

substitute x = -1, z = 2 in (1) eqn.

(-1) + y + 2(2) = 2

y -3 = 2

y = 5.

x  = -1, y = 5, z = 2.

Write a Fraction for the Point

Fraction:

Fraction is defined as an element of quotient field. Fraction can be represented as `x/y` where fraction variable 'x' denotes the value called as numerator and fraction variable 'y' denotes the value called as denominator and the denominator 'y' is not equal to zero. It is used to write the fraction format for the given point.

Thus the fraction is classified as follows,

• Simple fraction
• Proper fraction
• Improper fraction
• Complex fraction

write a fraction for the point : Types of fractions


Simple fraction:

Simple fraction is a fraction, which has both numerator and denominator as whole number.

Ex:

`1/5` , `2/7` , `8/9`

Proper fraction:

It is a fraction, which has a numerator less than its denominator, and the value of that fraction is less than one.

Ex:

`3/5` , `1/8` , `24/25`

Improper fraction:

Improper fraction is a fraction, where the top number of fraction that the numerator is greater than or equal to its own denominator (bottom number) and the value of that fraction is greater than or equal to one.

Ex:

`7/2` , `45/23` , `123/120`

Complex Fractions:

If a fraction of numerator and denominator contains a fraction, it is called complex fraction.

The complex fraction is also called as a rational expression because it has a numerator and denominator with fraction. Otherwise, the overall fraction includes at least one fraction.

Ex:

` (7/3) / (4/5)`


Example problems for write a fraction for the point:


Ex 1

Write the fraction for the following point: (0.5, 1.5, and 0.88)

Sol:

0.5

Step 1 :( multiply  and divide by 10 on both sides, we get )

`(0.5)*(10)/10`

= `5/10`

Step 2: simplifying we get

=`1/2`

like wise for the following numbers we get,

1.5 = `3/ 2 ` ( multiply and divide by 2 on both sides)

0.88 = `8/9 ` ( multiply and divide by 10 on both sides)

Ex: 2

Write the Equivalent fraction for the following points: (0.25, 0.75, 2.5, and 50)

Sol:

0.25 = `1/4 ` = `2/8` = `3/12 `

0.75 = ` 3/ 4` = `6/8` = `9/12`

2.5 = `5/2` = `10/4` = `15/6 `

50.0 = `100/2` = `200/4 `

study graph numbers

Number:

     A number is defined as a numerical thing, which is used for measuring and counting. It is also called as numeral and includes zero, negative numbers, rational numbers, irrational numbers, etc,. The procedure of numerical operation involves one or more numerical as input and generate its relevant numerical output. This operation includes arithmetic function such as addition, subtraction, multiplication, division, and exponentiation.

Study about graph numbers:

Composite Number:

A Composite Number is a number, which can be divided with evenly. The composite number has additional than two factors with one and itself.

Otherwise, Composite number can be defined as numeral (integer) that is accurately divided with minimum one factor except one and itself. Composite number has infinite numbers also. However, composite numbers are not prime numbers.

Prime Number:

A prime number is a number, which can be divided only with one and itself. The prime number has only two factors such that one and itself.

Cardinal numbers:

Cardinal number is defined as counting numbers in words, which is representing the quantity such as five dogs, three boys. Otherwise, it is referred to as overview of natural numbers, which is calculating the cardinal (size) of the given sets.

Ordinal number:

     Ordinal numbers are numbers, which denote the order or location of objects with number words in a series such as ‘first’, ‘second’, ‘third’... It is also called as ordinals.

Graphing numbers:

     Graphing numbers is the graphical representation of integers with inequalities in horizontal line. It is a visualizing result of number line with simple steps.

Inequality:

     Inequality is defined as two real numbers or two algebraic expressions are related with functioning a sign as ‘<’ (less than), ‘>’ (greater than), ‘≤’ (less than or equal) and ≥ (greater than or equal).

Examples for graph:

Example 1) graph the following points

A= 3 + 2i
B =  -4 + 5i
 C = -5 - 4i
 D = i

Solution: 

2) Graph the following inequalities:

a) -1≤w≤4

Solution:

The value of ‘w’ is -1, 0, 1, 2, 3, 4

So graphing of ‘w’ on number line is

 

b) 1≤q<5

Solution:

The value of ‘q’ is 1, 2, 3, 4

So graphing of ‘q’ on number line is

Study Exponentiation

In math, exponentiation is the operation, which is written as the form of an. Where a and n is said to be base and exponent and n is any positive integer. Normally, exponentiation means that repeated multiplication. Otherwise, exponentiation an is the product of n factors of a. The exponent is usually placed as a superscript to right of base value. We are having many properties for exponentiation. Let see properties and example problems for exponentiation.


Properties - Study Exponentiation


We are having seven number of exponentiation properties that used for solving problems with exponentiation. In this properties, a, m and n are any integer values.

Product of like bases:

am an = am+n

Quotient of like bases:

`a^(m)/a^(n)` = am-n

Power to a power:

` (a^(m))^(n)` = amn

Product to a power:

(ab)m = am bm

Quotient to a power:

`(a/b)^(n)` = `a^(n)/b^(n)`

Zero exponent:

a0 = 1

Negative exponent:

a-n = `1/a^(n)` or `1/a^(-n)` = an

These are the properties that are used for exponentiation problems in study math.


Example Problems - Study Exponentiation


Example 1:

Solve 23 22.

Solution:

Given, 23 22.

This is in the form of am an, so we need to use am an = am+n property.

Here, m = 3 and n = 2 and a = 2.

Thus, 23 22 = 23+2

= 25

= 2 × 2 × 2 × 2 × 2

= 32
Hence, the answer is 23 22 = 32.

Example 2:

Shorten the following `5^5/5^3`.

Solution:

Given, `5^5/5^3` .

This is in the form of `a^m/a^n` , so we need to use `a^m/a^n` = am-n property.

Here, m = 5 and n = 3 and a = 5.

Thus, `5^5/5^3` = 55-3

= 52

= 5 × 5

= 25
Hence, the answer is `5^5/5^3` = 25.

That’s all about the study exponentiation.

Solving Change of Base Formula

The solving change of base formula is known as formulas which it permits us to rework a logarithm by means of the logs that may be is written with different base.

The change of base formula is given by,

Log a x = log b x / log b a

Here, assume that a, b and x are positive where a≠1 and b≠1.

Importance in solving change of base formula:

• Using the change of base formula we can change any base to another base. The most commonly used bases are base 10 and base e.

Log a x = log b x / log b a

• The solving change of base formula is used highly if calculators to assess a log to several base further than 10 or e.
• At the solving change of base formula having the value of x which is superior than zero.
• The log of a number to a given base is the power or an exponent to which the base must be raised in order to produce that number.

Advantages  in using change of base formula:

• Change the numeral bases, like convert  from base 2 to base 10m which is known as base conversion.
• The logarithmic change-of-base formula is applicable regularly in algebra and calculus.
• It is used for varying among the polynomial and normal bases.

solving Examples using change of base formula:


1) Solve  log 816

Solution:

By solving the change of base formula

=> log a x = log b x / log b a

log 8 16 = log 2 16 / log 2 8

=  4 / 3

2) Solve log 918

Solution:

By solving the change of base formula

=> log a x= log b x / log b a

log 9 18 = log 2 18 / log 2 9

= 4.169 / 3.169

= 1.315

3) Solve log 2 5.

Solution:

By solving the change of base formula

=> log a x= log b x / log b

log 2 5 = log 10 5 / log 102

The approximate value of the above expression is solving by,

=0 .6989 / 0.30103

=0 .3494

Practice problems in solving change of base formula :

1) Solve log 3 9 using the change of base formula.

Answer: 2

2) Solve log 10 8 using the change of base formula

Answer: 0 .9030

learning probability and odds

The probability of an incident is a proportion that tells how possible. It is that an event will take place. The numerator is the number of favorable outcomes and the

Denominator is the number of possible outcomes.

Probability:   number of flattering outcomes / number of possible outcomes

Probability is we can try to measure the chances of it occurrence

Explanation of learning probability and odds


Here learning the probability and odds

Numerical measure of the likelihood of an event to occur. If an inspection there are n possible  ways  exhaustive and mutually exclusive and out of them in m ways in the event. A occurs, then the probability of occurrence of the event. A is given by P (a) = m/n

If  in a random  sequence  of n trials of an event, M are favorable  to the event , the  probability of that event occurring is the limit of the ratio M/n, when  n is very  large , this lies between 0 and 1

P (a) = 0 means that the event can not take place.

P (a) = 1 means the event is bound to occur.

The events, A2, A3……..An are said to the mutually exclusive if ,

P(Ai∩ Aj)=0

for i=1,2,3,……n.

J=1,2,3,…….n, I  j

He events are said to be exhaustive if

P(A1)+ P(A2)+ P(A3)_.......... P(An)=1 if ,

A ∩   B  0 then

P(Aυ B)=P(A)+ P(B)

And if  A∩  B   0v then

P(A υ B)=P(A)+ P(B)-P(A ∩ B)


Learning Example of probability and odds:


Learning the probability and odds problems

1. when you toss a coin, it can fall two ways. The probability of getting a head on one roll of a coin is one chance out of two.

Solution:

P(h) means the   probability of getting a head on one toss of a  coin.

P(t) means the   probability of getting a tail on one toss of a  coin.

Step 1:  the number of favorable outcomes for head =1

Step 2:  the number of favorable outcomes for tail =1

Step 3:  number of possible outcomes    =2

Step 4:  the probability of the event for head = number of favorable outcomes / number of possible outcomes

Step 5:  the probability of the event for head= P (h) =n (e)/n(s)

Step 6:   the probability of the event for head=1/2

Answer:  1/2

Factor Tree Online

Before learning factor tree online, we'll learn important terms related with factor tree. Those terms are given below:

Factor: A factor of a number is the exact divisor of that number. In other words, a factor of the number divides the number leaving remainder 0.

We know that 15 = 1 x 15 and 15 = 3 x 5. This shows that 1, 3, 5 and 15 exactly divides 15. Therefore, 1, 3, 5 and 15 are all factors of 15.

Prime Number: Each of the numbers which has exactly two factors, namely, 1 and itself, is called a Prime Number. For example: 2, 3, 5,11, 13 etc.

Prime Factor: A factor of a given number is called a prime factor if this factor is a prime number. A prime factor cannot be factored further.Example: 2 and 3 are prime factors of 12 while 4, 6 and 12 are not.

Prime factorization : To express a given number as a product of prime factors is called Prime Factorization or complete factorization of the given number.

Example: 36 = 2 x 2 x 3 x 3, 45 = 3 x 3 x 5 etc.

Factor Tree: A diagram that shows the prime factors of a number is known as Factor tree. A factor tree contains all the prime factors of a given number.


Process to make a factor tree online


Start with the number.

• Make branches of factors - numbers that multiply to give you the original number
• Keep reducing each factor to its lowest possible prime factors.
• When we have all the prime factors identified, re-arrange them from least to greatest.
• Re-write them as exponents.

Illustration: Lets see the factor tree online preparation of 36 and 180 in the figure below:

1) Factor tree of 36:                                                                       
2. Factor tree of 180




Uses Of Factor Tree online


1. Factor Tree online is used to find the GCF or HCF

Example: Find the greatest common factor of 18 and 54.using factor tree

18
3 x 6
3 x (2x3)
The prime factors of 18 are 2, 3, and 3.

54
6 x 9
(2x3) x (3x3)
The prime factors of 54 are 2, 3, 3, and 3.

The prime factors that 18 and 54 have in common are 2, 3 and 3, so the greatest common factor is 2 x 3 x 3 = 18.

2. Factor Tree is used to find the LCM or LCD

Example: Find the LCM of 12 and 36 using factor tree

Step 1: Find the Prime factors for

12 = 2 x 2 x 3
36 = 2 x 2 x 3 x 3

Step 2: Multiply each prime factor the greatest number of times it appears in any one factorization.
The most number of times 2 appears in either factorization is twice. The most number of times 3 appears in either factorization is twice.

Thus, LCM = 2 x 2 x 3 x 3 = 36