Factoring Polynomials using Algebra

The reverse process of multiplying polynomials is defined as factoring polynomials. Consider that when we factor a number, we are searching for prime factors that multiply together to give the number. When a polynomial is factorized, usually only the polynomials are broken down to have integer coefficients and constants. Simplest way for factoring is that a common factor for every term. So we can factor out the common factor in the polynomials.

For example,  8=4*2, or 16=4*4


Problems:


Factoring Polynomials using Algebra Tiles:

Example for Factoring Polynomials using Algebra Tiles 1:

Factorize the polynomial x3 – 5x2 – 12x + 36

Solution for Factoring Polynomials using Algebra Tiles 1:

Sum of the coefficients of terms: 1–5–12 + 36 = 18 ≠ 0. ∴ (x–1) is not a factor.

Sum of the coefficients of even degree terms = –5 + 36 = 31

Sum of the coefficients of odd degree terms = 1 – 12 = –11

Since they are not equal we guess that (x + 1) is also not a factor. Let us check whether x – 2

is a factor. By synthetic division method

                 2 | 1       -5     -12        +36

                    |

                    |          +2      -6          -36
                    ________________________

                      1        -3     -18     |    0      

                  _________________________



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

                      x2 – 3x – 18 = x2 – 6x + 3x – 18

                                       = x (x–6) + 3 (x–6) = (x + 3) (x – 6)

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


Sample problem

Example for Factoring Polynomials using Algebra Tiles 2:

Factorize 2x3 + x2 – 5x + 2

Solution for Factoring Polynomials using Algebra Tiles 2:

Since the sum of the coefficients of all the terms: 2 + 1 – 5 + 2 = 5 – 5 = 0

We guess that (x – 1) is a factor.

By synthetic division,

                              1 | 2         +2        -5        +2

                                |

                                |              2        +3        -2

                                 ________________________

                                     2         3         -2     |   0        

                                _________________________

                     Remainder is 0. Quotient is 2x2 + 3x – 2

To find other factors, factorize the quotient,

                   2x2 + 3x – 2 = 2x2 + 4x – x – 2

                                = 2x (x + 2) – 1 (x + 2) = (x + 2) (2x – 1)

                   ∴ 2x3 + x2 – 5x + 2 = (x – 1) (x + 2) (2x – 1)

Mean Median Average

Mean

The mean is the average of the numbers.

It is easy to evaluate: Just add up all the numbers, then divide by how many numbers there are.

Example:

what is the mean of 2, 7 and 9?

Solution: 2 + 7 + 9 = 18
= 18 ÷ 3

= 6
Mean is 6

Median


The middle number (in a sorted list of numbers). Half the numbers in the listing are less, and half the numbers are greater are called as the median.

To find the Median, place the numbers you are given in value arrange and find the middle number.

If there are two middle numbers then average those two numbers.

Average

Average - The middle or most general in a set of data. There are three types of standard in mathematics - the mean, the median and the mode.


Concept of mean median average


Average

The average is a calculated "central" value or rate of a set of numbers.

It is easy to calculate: add up all the numbers and divide by how many numbers there are and you will have the average.

Example:

the average of 4, 6 and 11

Solution:   (4+6+11)/3

= 21/3

= 7

Average is 7

Mean

The most general expression for the mean of a statistical distribution with a discrete random variable is the mathematical average of all the terms. To compute it, add up the values of all the terms and then divide by the number of terms. This expression is also called the arithmetic mean.

Median

The median of a distribution with a discrete random or chance variable depends on whether the number of terms in the distribution is even or odd. If the number of conditions is odd, then the median is the value of the term in the middle. This is the value such that the number of conditions having values greater than or equal to it is the same as the number of terms having values less than or equal to it. If the number of conditions is even, then the median is the average of the two terms in the middle, such that the number of terms having values greater than or equal to it is the same as the number of terms having values less than or equal to it.


Example of mean median average


A student has gotten the subsequent grades on his tests: 87, 95, 76, and 88. He wants an 85 or better overall. What is the least amount of grade he must get on the last test in order to achieve that average?

The unknown score is "x". Then the desired average is:

(87 + 95 + 76 + 88 + x) ÷ 5 = 85

Multiplying through by 5 and simplifying, I get:

87 + 95 + 76 + 88 + x = 425
346 + x = 425
x = 79

He needs to get at least a 79 on the last test.

The Binomial Distribution

Binomial Distribution is a statistical experiment which means the number of successes in n repeated trials of a binomial experiment. It is also called as Bernoulli distribution or Bernoulli trial.

For example:

For a clinical trial, a patient may live or die. Here the researcher faces the number of survivors and not how much time the patient lives after treatment.


Properties and Formula for binomial distribution


For example:

For a clinical trial, a patient may live or die. Here the researcher faces the number of survivors and not how much time the patient lives after treatment.

We take a coin and flipped two times. Here we calculate the count of number of heads(successes). So the binomial distribution is

Number of heads          Probability

No head                             0.25

One head                           0.5

Two head                           0.25

Properties of Binomial Distribution

The experiment  has n repeated trials.

Each trial can have two possible outcomes. One is success and another one is failure.

Here the trials are independent.

Mean = n * P.
Variance = n * P * (1 – P).
Standard Deviation  =  sqrt[ n * P * ( 1 – P ) ].


Binomial distribution Formula

b(x; n, P) = nCx * Px * (1 - P)n – x

Here the Notation are,

B(x; n, P)   =  Binomial Probability.

X   =  successes

N   =  number of trials

P    =  Probability of success

nCx  = Number of combinations of n trials, x is success.


Example Problem(the binomial distribution)


A die is tossed 6 times. What is the Probability of getting exactly 2 fours?

Solution

Here n = 6, x = 2,  probability of success on a single trial = 1/ 6 or 01.167.

Therefore, The binomial probability is,

b( 2; 6, 0.167 )             =  6C2 * ( 0.167 )2 * ( 1 – 0.167)6 – 2

=  ( 6! / 2! * (6-2)!) * 0.0279 * ( 0.833)4

=  (6! / 2! * 4!) * 0.0279 * 0.481

= 15 * 0.0279 * 0.481

b( 2; 6, 0.167 )             = 0.201. Answer.

Cumulative Binomial probability

It  refers to the binomial probability falls within a specified range that is greater than or equal to a mentioned lower limit and less than or equal to a mentioned upper limit.

For example

Cumulative binomial probability of obtaining 5 or fewer heads in 10 times of a coin.

b( x <= 5; 10, 0.5)=   b( x = 0; 10, 0.5) + b( x = 1; 10, 0.5) +…… + b ( x = 5; 10, 0.5)

Rational Equations Solving Online

Learning equation of the form P(x)/Q(x) over the set of real numbers and Q(x) ≠ 0 where P(x) and Q(x) are two polynomials is called rational equation. Rational equations made easy for learning through online.

Example for rational equations: 3/8 = 3/(d + 4)

(x4 + x3 + x + 1)/(x + 5) = 9/5

1/(x + 9) = 3

12a-6 = 3a-2

Example problem for rational equations solving online:


The following example problems give idea for solving rational equations.

Example 1:

Find x value of the rational equation (6x + 36) / (9x + 54) = 6x

Solution:

Step 1: Given equation

(6x + 36) / (9x + 54) = 6x

Step 2: Take 6 and 9 out as common term from numerator and denominator respectively.

6(x + 6) / 9(x + 6) = 6x

Step 3: Cancel the term (x + 6)

6/9 = 6x

Step 4: Rearrange the above equation,

6/(6 * 9) = x

1/9 = x

x = 1/9

Example 2:

Find out x value for the equation x/(x - 4) + (1/x - 8) = 6/(x2 - 12x + 32)

Solution:

Step 1: Given equation

x/(x - 4) + (1/x - 8) = 6/(x2 - 12x + 32)

Step 2: Replace the term (x2 - 12x + 32) by (x - 4) (x - 8)

x/(x - 4) + (1/x - 8) = 6/ (x - 4) (x - 8)

Step 3: Make common denominator.

(x/(x - 4))((x - 8)/(x - 8)) + ((1/x - 8))((x - 4)/(x -4)) = 6/ (x - 4) (x - 8)

(x2 - 8x) / (x - 4)(x - 8) + (x - 4) / (x - 4)(x - 8) =  6/ (x - 4) (x - 8)

Step 4: Cancel the term (x - 4)(x - 8), we get

(x2 - 8x) + (x - 4) = 6

Step 5: Rearrange the above equation

x2 - 7x - 4 = 6

x2 - 7x - 4 - 6 = 0

x2 - 7x - 10 = 0

Step 6: On factorizing, we get

x = 8.217, - 1.217

This is how the rational equations can be solved through online.


Homework problem for rational equations solving online:


A few homework problems are given below  for solving rational equations.

1) Simplify and find x for the equation (4x + 12)/(4x + 32) = 1/(x + 8)

2) Find x value for the equation 1/(x + 7) = 4

3) Solve and find x for the equation (x2 - 12x + 32)/(x - 8) = 8

Solutions:

1) x = -2

2) x = -5

3) x = 12

Improper Integrals Solve Online

In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.

Specifically, an improper integral is a limit of the form


In which one takes a limit in one or the other (or sometimes both) endpoints. (Source: Wikipedia)


Example problems for solve online improper integrals


Online improper integrals example 1:

Solve:

Find the value of the integration function

`lim_(b->0) int_3^b(2x + 3)dx`

Solution:

Integrate the given function with respect to x, we get

= `lim_(b->0)` [2 `(x^2 / 2)` + 3x]b3

Substitute the lower and upper limits, we get

=  `lim_(b->0)` (b2 - 3b) - (9 + 9)

Substituting the value of b, we get

= (0) - 18

After substituting the limits, we get

= - 18

Answer:

The final answer is - 18

Online improper integrals example 2:

Solve:

Find the value of the integration function

`lim_(b->2) int_0^b(7x^3 + 3x^2)dx`

Solution:

Integrate the given function with respect to x, we get

= `lim_(b->2)` [7 `(x^4 / 4)` + 3`(x^3 / 3)` ]b0

Substitute the lower and upper limits, we get

=  `lim_(b->2)` [`(7 / 4)` b4 + b3] - (0)

Substituting the value of b, we get

= [(7 / 4) 24 + 23] - 0

After substituting the limits, we get

= 36

Answer:

The final answer is 36

Online improper integrals example 3:

Solve:

Find the value of the integration function

`lim_(b->1) int_2^b(6x^2 + 18x)dx`

Solution:
Integrate the given function with respect to x, we get

= `lim_(b->1)` [6 `(x^3 / 3)` + 18 `(x^2 / 2)` ]b2

Substitute the lower and upper limits, we get

=  `lim_(b->1)` (2b3 + 9b2) - (16 + 36)

Substituting the value of b, we get

= (11) - 52

After substituting the limits, we get

= - 41

Answer:

The final answer is - 41


Practice problems for solve online improper integrals


Online improper integrals example 1:

Solve:

Find the value of integration of the function

`lim_(b->0) int_5^b(6x)dx`

Answer:

The final answer is - 75

Online improper integrals example 2:

Solve:

Find the value of integration of the function

`lim_(b->6) int_0^b(12x + 2)dx`

Answer:

The final answer is 228

Mean Statistics Definition

Introduction:

Let us see the introduction of mean statistics. In statistics the mean is a mathematical average of a set of numbers. The average of the mean statistics is dividing the total by the number of scores and calculated by adding up two or more scores. Mean Statistics like as many other sciences of a developing discipline. In statistics has been defined in different times and different manners. We discuss the definitions of mean statistics.


Mean Definition:

The definition of the mean in mathematics, the statistical discrete random variable average of all the terms. A finite set of terms are forms rarely used in statistics are other expressions for the mean. The lowercase Greek letter mu (µ) is the expected value. The average value of mean statistics is a numerical set. The number of members in the group of numbers is by dividing the sum of a set of numbers.

Statistics Definition:

Statistics is definition of the technique or the scientific method is used to analyzing, collecting, interpreting, classifying, and data. The statistics is used to obtain the analyses, summaries, compare and present the numerical data. The statistics definitions are a science which deals with the application and investigates the statistical significance. 

These are the definitions of mean statistics.

Examples:

1. Find the mean of the following numbers 2, 5, 23, 15, 15, and 6.

Solution:

The given numbers are 2, 5, 23, 15, 15, and 6.

The average of set of numbers =2+5+23+15+15+6 / 6

= 36 / 6

=6

Answer: Mean is 6

2. Find the mean of the following numbers 5, 7, 34, 56, 23, 46, 56, and 12.

Solution:

The given numbers are 5, 7, 34, 56, 23, 46, 56, and 12.

The average of set of numbers =5+7+34+56+23+46+56+12/8

=   239 / 8

=29.88

Answer:

Mean is 29.8

3. Find the mean of the following numbers 56, 34, 23,  and 15.

Solution:

The given numbers are 56, 34, 23,  and 15.

The average of set of numbers=56+34+23+15/4

= 128 / 4

=32

Answer: Mean is 32

These are the example problems for definition of mean.

Number line

Learn on number line in this page and gain quality algebra help. First a brief introduction is given on the whole concept of number line and further the topic is explained.

Mathematically, a number line is referred to as an image of a straight line, which has several points on it as real numbers. Those points are indicated by integers. Particularly marked points with even spaces expose them on the line. Thus, real numbers are represented in each direction. Number line has both positive number and negative number held at correct points on line, in that zero is the center point of the number line, while right side of zero is positive number and left side of zero is negative number.



Numbers on a line can be represented horizontally as well as vertically. Normally, number line is represented horizontally.


Steps to draw


The following  are the steps to draw the number line-

• Step 1: Draw a horizontal straight line. Because mostly the number line is represented as horizontal line
• Step 2: Draw the arrow on both ends of number line.
• Step 3: Point the origin zero on the number line.
• Step 4: Write positive integer on the right side of origin with even spaces.
• Step 5: Write negative integer on the left side of origin with even spaces.
• Step 6: Mark all integers over the number line.
• Step 7: Plot the answers for given question.

Points on a number line


Ex 1: There are three persons on the origin namely as A, B, C. A walks backwards 2 points; B & C walks towards 4 points and 3 points respectively. Point out their current position.

Sol:




Points on a number line in decimal form

Case 1: Positive Decimal on number line

Let take the positive decimal as 3.75 and points it as in number line.

Generally, we know that decimal have two section,

• before the decimal point
• after the decimal point.

When make the decimal number on number line,

Step 1: keep the number before the decimal point. Here the 3 is the positive number before the decimal point.

Step 2: Now after the positive number three on a number line, we count the number in number line in between 3 and four as per the number have after the decimal point.

Step 3: Now we get the correct points for the given decimal as 3.75

Case 2: Negative decimal number on simple number line

Let take the number as -6.5 and mark it on the number line.



Step 1: keep the number before the decimal point. Here the 6 is the negative number before the decimal point.

Step 2: Now after the negative number 6 on a number line, we count the negative number in number line in between 6 and 7 as per the number have after the decimal point.