Polynomials Algebra Tiles

In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number. In this article we shall discuss about polynomials algebra tiles. (Source: Wikipedia)


Sample Problem for Polynomials Algebra Tiles:

Solved polynomials algebra tiles problem:

Example 1:

Factorize the polynomial 2x3 – 10x2 – 24x+ 72

Solution

Sum of the coefficients of terms: 2–10–24 + 72 = 40 ≠ 0.

Therefore the value (x–1) is not a factor.

Sum of the coefficients of even degree terms = –10 + 72 = 62

Sum of the coefficients of odd degree terms = 2 – 24 = –22

Since they are not equivalent we estimate that (x + 1) is also not a factor. Let us check whether x – 2 is a factor. By synthetic division method

2 | 2       -10     -24        +72

|

|          +4      -12          -72
________________________

2        -6     -36     |    0

_________________________

Since the remainder is 0, (x – 2) is a factor. To find other factors

2x2 – 6x – 36 = 2x2 – 12 x + 6x –36

= 2x(x –6) + 6(x–6) = (2x + 6) (x – 6)

Therefore, x3 – 5x2 – 12x + 36 = (x–2) (x–6) (x+3)


Example 2:

Factorize 4x3 + 2x2 – 10x + 4

Solution:

Since the addition of the coefficients of all the terms: 4 + 2 – 10 + 4 = 10 – 10 = 0

We guess that (x – 1) is a factor.

By synthetic division,

1 | 4         +2        -10        +4

|

|              4        +6        -4

________________________

4         6         -4     |   0

_________________________

Remainder is 0. Quotient is 4x2 + 6x – 4

To find other factors, factorize the quotient,

4x2 + 6x – 4 = 4x2 + 8x – 2x – 4

= 4x (x + 2) – 2 (x + 2)

= (x + 2) (4x – 2)

∴ 2x3 + x2 – 5x + 2 = (x – 1) (x + 2) (2x – 1)
Practice Problem for Polynomials Algebra Tiles:

Factorize 2x2 + 3x -2

Answer: (x + 2) (2x – 1)

Factorize X2 – 3x – 18

Answer: (x + 3) (x – 6)

Add and Subtract Fractions Calculator

Add and subtract calculator which is used for finding the addition and subtraction for various value of a fraction. Calculator is a process of calculating fractions with different values.In general fraction contains two parts numerator and denominator. the upper part is the numerator and lower part is a denominator which depends upon the this two parts. there are types of fractions such as proper,improper fraction and mixed fraction.

Rules for Add and Subtract on Fractions Calculator:

fraction calculator

First choose the operations in the column shown(add,subtract,multiply and divide)
then,fraction #1 enter the fraction value and in the fraction#2 then press the calculate
We get final result in column below the calculate.
For the next operation press reset and follow the steps.

Fraction shown above will be the calculator for the fraction addition and subtraction steps in the calculator will be explained below.
Addition on fractions:

For adding same denominator fractions just add all the numerator and keep same denominator.


For addition with different denominators fraction:

Find the l.c.m. for all the denominators given .
Change into equivalent fraction with the same LCM denominator
By taking  the LCM common in  the denominator and add all the numerators.


Subtraction on fractions :

For  subtract same denominator fractions subtracting  all numerator and keep same denominator.


For subtraction with different  denominator:

Finding  l.c.m. for all the denominators given.
Change equivalent fraction with same l.c.m denominator and then subtract


Problems for Add and Subtract Calculator:
Example 1:
Add   1/4 and 2/4

Solution:
here we have the same denominator so, just add the numerator keep the denominator same.
1/4 +2/4 = (1+2) /4

=3/4

Example 2:
Add  4/5  and 1/3

Solution :
The LCM of 5 and 3 is 15.
Therefore,4/5+1/3 = (4xx3)/(5xx3)+(1xx5)/(3xx5)

=12/15+5/15

=17/15

Example 3:
Add 2 4/5 and 3 5/6

Solution:
The given fraction is a mixed fraction first change into proper fraction and add
 2 4/5 +3 5/6=(2xx5)+4/5+(3xx6)+5/6=14/5+23/6

so, the denominator is different LCM of 5,6 =30
=(14xx6)/(5xx6)+(23xx5)/(6xx5)

=84/30 +115/30

=119/30

Example 4:
Subtract  3/5 -1/5

Solution:
here, same denominator just subtract numerator and keep denominator is same.

 3/5-1/5 =(3-1)/5
=2/5
Example 5:
subtract 4 2/5-2 1/5

Solution :
The given fraction is a mixed fraction first change into proper fraction and subtract
(4xx5)+1/5 -(2xx5)+1/5

so, 21/5 -11/5

=(21-11)/5

=10/5

Example 6:
Subtract  3/4   from  5/6

Solution :
We need to find equivalent fractions of 3/4 and 5/6 , which have the same denominator.
This denominator is given by the LCM of 4 and 6. The required LCM is 12.

Therefore, 5/6-3/4 =(5xx2)/(6xx2)-(3xx3)/(4xx3)
=10/12-9/12

=1/12

Definition of Subset Learning

We come across set of different kinds in everyday life. A football team is set of players, a class is a set of students, a set of books in mathematics and so on. Basically, the set is an undefined term.  Anyhow, it can be a well defined  collection of objects. Here the word 'objects' has been taken in wider and broader sense. Mathematically, the numbers, words, letters, signs, symbols, thoughts, etc. all are the objects. Let us learn about a set and  definition of a subset in this article.

Learning Definition of a Subset

Definition

If each element of A is an element of B, then the set A is said to be a sub set of the set B.

Symbolically, we represent this by A `sub` B and read it as ' A is a subset of B'.

Thus A `sub` B `hArr` (  x `in` A   `rArr` x `in` B ).

If A `sub` B and A `!=` B then the set A is said to a proper subset of B.

learning examples of subset

1. {1, 2, 3, 4, 5 } `sub` { x : x is a natural number }

2. { a, b, c, d } `sub` { a, b, c, d, e}

3. Set of all vowels is a subset of set of all alphabets

4. A = { set of all integers}

B = { set of all positive integers}

B is a subset of A.

5. T = { x : x is a student of class 9 in your school }

W = { x : x is a student in your school}

T is a subset of W.



Definition of Subset Learning - Points to Remember

1. `phi` `sub` A ( since `phi` has no elements, it is a subset of every set)

2. A `sube` A ( every set is a subset of itself)

3. A `sube` B and B `sube` A `hArr`    A = B

4. A `sub` B and B `sub` D `=>`   A `sub` D

5. If A `sub` B, then x `!in` B `rArr` x `!in` A.

Adding Exponents Worksheet

Exponents are nothing but multiplying the number by itself.  For example, y2 is the same as y * y. Exponent denotes the number of times the number needed to be multiplied by itself. An exponent is nothing but a superscript, or small number that can be written at the top right edge of a number, variable, or set of parentheses. In this article we shall discuss the adding exponents worksheet.

Problems for Adding Exponents Worksheet:

Adding exponents worksheet:

Problem 1:

To find x3y4. x5y3

Solution:

Rearrange the factors and multiply the exponent terms, using the rule

Here,

am.an = am+n

= (x3. x5)( y3. y4)

= x8. Y7

X and y are different terms,

So the final result is x8. Y7

Problem 2 for Adding exponents worksheet:

To find  (32)3

Solution:

(32)3 = 93 =  729 using power rule,

The answer is 729

Problem 3:

To find  (22)3

Solution:

(22)3 =(2x2)3 =  43=64 using power rule,

The answer is 64

Mixed variables for multiply exponents worksheet:

Have a mix of variables:

Example 1:

=xy2z y3z

=x y2+3z1+1

=xy5z2

so the result is =x3y5z2

Example 2:

=x3y3z3 .  x2yz

=x3+2 y3+1 z3+1

=x5y4z4

so the result is =x5y4z4

With constant examples:

Example 1:

=5xyz .4xyz

=(5 4)x1+1 y1+1 z1+1

=20x2y2z2

The result is =20x2y2z2

Example 2:

=7xyz.3 x2yz

=(73 ) x1+2y1+1z1+1

=21x3 y2z2

The result is  =21x3 y2z2
More Examples for Multiply Exponents Worksheet:

Example 1:

=5xyz `xx` 4xyz

= (5`*` 5)x1+1 y1+1 z1+1

=25x2y2z2

The result is =25x2y2z2

Example 2:

=7xyz `xx` 4 x2yz

= (7`*` 4) x1+2y1+1z1+1

=28x3 y2z2

The result is  =28x3 y2z2

Example 3:

=5xyz `xx` 7xyz

=(7`*` 5)x1+1 y1+1 z1+1

=35x2y2z2

The result is =35x2y2z2

Example 4:

=7xyz  ` xx` 2 x2yz

= (7`*` 2 ) x1+2y1+1z1+1

=14x3 y2z2

The result is  =14x3 y2z2

Define Positive Integer

The definition of positive integer is one of the most important topic in mathematics. Positive integer is always present after the zero in the number line. The symbol used for representing the positive integer is called as the " + ". All the mathematical operations are done using the positive integer. In this article, we are going to see about the positive integer with brief explanation and some example problems.

Explanation to Define Positive Integer
The explanation for to define the positive integer is given below the following,

Addition problem for defining the positive integer
Subtraction problem for defining the positive integer
Multiplication problem for defining the positive integer
Division problem for defining the positive integer

Example Problems to Define Positive Integer

Example 1: Add the following positive integer, 36 and 24

Solution:

36 and 24

3 6

2 4

___

6 0

___

Result: Positive integer addition = 6 0

Thus, this is the result to define the addition problem problem of positive integer.

Example 2: Subtract the following to define positive integer, 36 and 24

Solution:

36 and 24

3 6

2 4

___

1 2

___

Result: Positive integer subtraction = 1 2

Thus, this is the result to define the subtraction problem of positive integer.

Example 3: Multiply the following to define positive integer, 36 and 24

Solution:

36 and 24

3 6

2 4

____

8 6 4

____

Result: Positive integer multiplication = 6 0

Thus, this is the result to define the multiplication problem of positive integer.


Example 4: Divide the following to define positive integer, 48 and 6

Solution:

48 and 6

6 ) 4 8 ( 8

4 8

____

0

____

Result: Positive integer division = 6 0

Thus, this is the result to define the division problem of positive integer.
Practice Problems to Define Positive Integer

Example 1: Add the following positive integer, 42 and 14.

Answer: 56

Example 2: Subtract the following positive integer, 42 and 14.

Answer: 28

Example 3: Multiply the following  positive integer, 42 and 14.

Answer: 588

Example 4: Divide the following positive integer, 56 and 7.

Answer: 6

Probability Distribution Definition

The word probability and chance are familiar to everyone. Many a time we come across statements like “It is possible that our school students may get state ranks in forthcoming public examination.

“Probably it may rain today”

Definition: The word chance, possible, probably, likely etc. convey some sense of uncertainty about the occurrence of some events. Our entire world is filled with uncertainty. We make decisions affected by uncertainty virtually every day. In order to think about and measure uncertainty, we turn to a branch of mathematics is called as probability.

Probability Distribution Definition-classical Definitions

Definition: If there are n exhaustive, likewise exclusive and in the same way likely outcomes of an experiment and m of them are favorable to an event A, and then the mathematical probability of A is defined as the ratio m/n.

Definition for random variable:

The outcomes of an experiment are represented by a random variable if these outcomes are numerical or if real numbers can be assigned to them. For example, in a die rolling experiment, the corresponding random variable is represented by the set of outcomes {1, 2, 3, 4, 5, 6} ; while in the coin tossing experiment the outcomes head (H) or tail (T) can be represented as a random variable by assuming 0 to T and 1 to H.

Types of Random variables:

(1) Discrete Random variable
(2) Continuous Random variable


Definition for Discrete Random Variable: If a random variable takes only a finite or a countable number of values, it is called a discrete random variable.

Example:
1. The number of heads obtained when two coins are tossed is a discrete random variable as X assumes the values 0, 1 or 2  which form a countable set.
2. Number of Aces when ten cards are drawn from a well shuffled pack of 52 cards.
Probability Distribution Definition-theoretical Distributions:

Theoretical probability distributions is classified into

1. Binomial Distribution

2. Poisson Distribution

3. Normal Distribution

4. Exponential Distribution

Multiplication Of Exponents

Exponents are significant in scientific notation, when large or small quantities are denoted as powers of 10. exponents are mentioned by superscripts, as in the examples above. But it is not always possible way to write them this method.  If x is the exponent to which is a minimum base quantity a is increased value, then a x can be written in ASCII as a power of x. In a scientific notation, the higher case letter E can be used to point out that a number is raised up to a positive or negative power of 10. For model take 125x3. Here 125 is coefficient of variable 'x’ Then 3 is the exponent value of x . Exponent value also known as power value.
Suitable Examples for Multiply Exponents


Exponents of 1 and 0
If the exponent is 1, then only have the variable itself (example a1 =a)

Generally need not to write the "1", but it sometimes helps to remember that y is also a1
Exponent of  0

If the exponent is 0, then the values are not multiplying by anything and the result is just "1" (example a0 = 1)
Multiplying Variables with Exponents

multiply this variable with exponents:

(z2)(z3)

So that z2 = zz, and z3 = zzz so that all the multiplies,

(z2)(z3)

= zzzzz

That is 5 "z"s multiplied mutually so the new exponent must be 5:

(z2)(z3)

= z5

The exponents say to that there are two "z"s multiplied by 3 "z"s for a total of 5 "z"s:

(z2)(z3)



= z2+3 =z5

So, the simplest method is to just add the components
Mixed Variables for Multiply Exponents:

Have a mix of variables:

Example 1:

=xy2z y3z

=x y2+3z1+1

=xy5z2

so the result is =x3y5z2    

Example 2:

=x3y3z3   x2yz

=x3+2 y3+1 z3+1

=x5y4z4

so the result is =x5y4z4

With constant examples:

Example 1:

=5xyz 4xyz

=(5 4)x1+1 y1+1 z1+1

=20x2y2z2

The result is =20x2y2z2

Example 2:

=7xyz 3 x2yz

=(73 ) x1+2y1+1z1+1

=21x3 y2z2

The result is  =21x3 y2z2