Factorization
The following are the different methods for factorization of a polynomials.
Taking Out Common Factor
If every term of a polynomial has a common factor then we divide each term by the common factor and multiply it by the remaining.
Factories: \[7{{(a+b)}^{2}}-14(a+b)\]
(a)\[(a+b)]7(a+b)-14]\]
(b) \[7[{{(a+b)}^{2}}-2(a+b)]\] (c) \[7(a+b)(a+b-2)\]
(d) \[7(a+b)(a+b-2a)\]
(e) None of these
Answer: (c)
Explanation:
Here, 7 and \[(a+b)\] is a common factor. Therefore, \[7{{(a+b)}^{2}}-14(a+b)=7(a+b)(a+b-2)\]
By Grouping
In this method, we group a given expression in such a way that we have a common factor. This method is effective for the expression in which it is not possible to take out common factor directly.
Factories: \[{{a}^{2}}{{y}^{2}}+(a{{y}^{2}}+1)y+a\]
(a) \[({{a}^{2}}y+1)(ay+1)\]
(b) \[(ay+y)({{a}^{3}}y+1)\]
(c) \[({{a}^{2}}y+1)(a+y)\]
(d) \[({{a}^{2}}{{y}^{2}}+a)(a+y)\]
(e) None of these
Answer: (c)
Explanation:
\[{{a}^{2}}{{y}^{2}}+(a{{y}^{2}}+1)y+a={{a}^{2}}{{y}^{2}}+a{{y}^{3}}+y+a=\]\[{{a}^{2}}{{y}^{2}}+a+a{{y}^{3}}+y\] \[=a(a{{y}^{2}}+1)+y(a{{y}^{2}}+1)+(a+y)\]
By Making a Trinomial of a Perfect Square
Adjust the trinomial in such a way that there must be a common factor.
The factor of \[{{\left( 3x-\frac{1}{y} \right)}^{2}}-8\left( 3x-\frac{1}{y} \right)+16+\left( z+\frac{1}{y}-2x \right)\left( 3x-\frac{1}{y}-4 \right)\]
(a)\[\left( 3x-\frac{1}{y}-4 \right)(x+z-4)\]
(b) \[\left( 3x-\frac{1}{y}-4 \right)(3x+z-4)\]
(c) \[\left( 3x-\frac{1}{y}-4 \right)(2x+3z-4)\]
(d) \[\left( 3x-\frac{1}{y}-4 \right)(x+3z-4)\]
(e) None of these
Answer: (a)
Explanation:
\[{{\left( 3x-\frac{1}{y} \right)}^{2}}-8\left( 3x-\frac{1}{y} \right)+16+\left( z+\frac{1}{y}-2x \right)\left( 3x-\frac{1}{y}-4 \right)\]
\[={{\left( 3x-\frac{1}{y} \right)}^{2}}-2\times 4\left( 3x-\frac{1}{y} \right)+{{4}^{2}}+\left( z+\frac{1}{y}-2x \right)\left( 3x-\frac{1}{y}-4 \right)\]
\[={{\left( 3x-\frac{1}{y}-4 \right)}^{2}}+\left( z+\frac{1}{y}-2x \right)+\left( 3x-\frac{1}{y}-4 \right)\]
\[=\left( 3x-\frac{1}{y}-4 \right)\left[ 3x-\frac{\bcancel{1}}{\bcancel{y}}-4+z+\frac{\bcancel{1}}{\bcancel{y}}-2x \right]\]
\[=\left( 3x-\frac{1}{y}-4 \right)(x-4+z)\]
By Using Different Formulae
By the inspection of given expression use suitable formula (Identifies) and factories it.
The factor of \[{{y}^{6}}+4{{y}^{2}}-1\] is
(a) \[({{y}^{2}}+y-1)({{y}^{4}}-{{y}^{3}}+2{{y}^{2}}+y+1)\]
(b) \[({{y}^{2}}+y-1)({{y}^{4}}-{{y}^{3}}+y+1)\]
(c) \[(y+{{y}^{3}}-1)({{y}^{4}}-{{y}^{3}}+{{y}^{2}}+y+1)\]
(d) \[({{y}^{2}}+y-1)({{y}^{4}}-{{y}^{3}}+{{y}^{2}}+y-1)\]
(e) None of these
Answer: (a)
Explanation:
\[{{y}^{6}}+4{{y}^{3}}-1={{y}^{6}}+{{y}^{3}}+3{{y}^{3}}-1)={{y}^{6}}+{{y}^{3}}+{{(-1)}^{3}}+3{{y}^{3}}\]
\[={{({{y}^{2}})}^{3}}+({{y}^{3}})+{{(-1)}^{3}}-3.({{y}^{2}}).y(-1)\]
\[=({{y}^{2}}+y-1)[{{({{y}^{2}})}^{2}}+{{y}^{2}}+{{(-1)}^{2}}-{{y}^{2}}.y-y(-1)-(-1)\times {{y}^{2}}\]
[Using Formula:\[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=(a+b+c)\] \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca)]\]
\[=({{y}^{2}}+y-1)({{y}^{4}}+{{y}^{2}}-1-{{y}^{3}}+y+{{y}^{2}})\] \[=({{y}^{2}}+y-1)({{y}^{4}}+{{y}^{3}}+2{{y}^{2}}+y-1)\]
Factorization of Quadratic Polynomial
The general form of a quadratic polynomial is \[a{{x}^{2}}+bx+c\], where \[a\ne 0\], such expressions are factories as following:
Step 1: Take "a" common from whole expression, i.e.\[a\left( {{x}^{2}}+\frac{b}{a}x+\frac{c}{a} \right)\]
Step 2: Factories the expression \[a\left( {{x}^{2}}+\frac{b}{a}x+\frac{c}{a} \right)\] by converting it as the different of two squares.
The factor of \[-10{{a}^{2}}+31a-24\] is______.
(a) \[\text{(2a}-\text{3})(\text{5a}-\text{8)}\]
(b) \[\text{(3}-\text{2a})(\text{8}-\text{5a)}\]
(c) \[\text{(2a}+\text{3})(\text{5a}-\text{8})\]
(d) \[\text{-(2a}-\text{3})(\text{5a}-\text{8})\]
(e) None of these
Answer: (d)
Explanation:
The given expression is \[-\text{1}0{{a}^{\text{2}}}+\text{31a}-\text{24}\]
Step 1: Taking -10, common from whole expression then the expression will be\[-10\left( {{a}^{2}}-\frac{31}{10}a+\frac{24}{10} \right)\]
Step 2: \[-10\left( {{a}^{2}}-\frac{31}{10}a+\frac{24}{10} \right)\]
\[=-10\left[ {{a}^{2}}-2.\frac{31}{20}a+{{\left( \frac{31}{20} \right)}^{2}}-{{\left( \frac{31}{20} \right)}^{2}}+\frac{24}{10} \right]\]
\[=-10\left[ {{a}^{2}}-2.\left( more... 
Types of Polynomials
We can classify polynomials by two methods
(a) According to degree
(b) According to terms
According to Degree
According to degree, polynomial can be classified into following types:
(i) \[x+y+z=9\] is a linear polynomial
(ii) \[3x+8\] is also a linear polynomial
(ii) \[{{(x-2)}^{2}}-{{(x-3)}^{2}}\] is a linear polynomial [Because\[{{(x-2)}^{2}}-{{(x-3)}^{2}}=2x-2\]]
Quadratic Polynomial
A polynomial in which the highest power of a variable is 2. The general form of quadratic polynomial is \[a{{x}^{3}}+b{{x}^{2}}+c;\,a\ne 0\].
(i) \[{{x}^{2}}+4x+3\] is a quadratic polynomial
(ii) \[{{(x+2)}^{2}}+{{(x-2)}^{2}}\] is a quadratic polynomial
Cubic Polynomial
A polynomial in which the highest power of variable is 3. The general form of cubic polynomial is \[a{{x}^{3}}+b{{x}^{2}}+cx+d\], where \[a\ne 0\].
(i) \[4{{x}^{3}}+3{{x}^{2}}+4x+5\] is a cubic polynomial
(ii) \[{{(4x+3)}^{3}}\] is a cubic polynomial
Biquadratic Polynomial
A polynomial in which highest power of variable is 4. The general form of biquadratic polynomial is \[a{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+5\].
(i) \[19{{x}^{4}}+16{{x}^{3}}+13{{x}^{2}}+14x+5\] is a biquadratic polynomial
(ii) \[{{(x+2)}^{4}}\] is a biquadratic polynomial
According to Number of Terms
According to number of terms, polynomials can be classified into following types.
(i) \[{{x}^{2}}yz\] is a monomial
(ii) \[\frac{1}{3}xyz\] is a monomial
Binomials
A polynomial which has only two terms is known as binomials.
(i) \[4{{x}^{2}}y+4z{{y}^{2}}\] is a binomial
Trinomials
A polynomial which has only three terms is known as trinomials.
(i) \[4{{x}^{3}}+4{{x}^{2}}+3x\] is a trinomial
Remainder Theorem
When \[P(x)\] is divided by \[(x-a)\],\[P(a)\] is the remainder, where \[P(x)\] is a polynomial of degree n > 1 and "a" is any real number.
Factor Theorem
\[(x-a)\] is said to be factor of \[p(x)\] if and only if \[p(a)=0\], where \[p(x)\] is a polynomial of degree\[n\ge 1\] and a is any real number.
Find the remainder when \[\text{p(y)}=\text{12}{{\text{y}}^{\text{3}}}-\text{13}{{\text{y}}^{\text{2}}}-\]\[\text{6y}+\text{7}\] is divided by \[\text{3y}+\text{4}\].
(a) \[36\frac{5}{9}\]
(b) \[-36\frac{5}{9}\]
(c) 36
(d) \[-\frac{829}{9}\]
(e) None of these
Answer: (b)
Explanation:
\[3y+4=0\] \[\Rightarrow \] \[y=-\frac{4}{3}\]
By more... 
Introduction
Previously we have studied about algebraic expression and various operations. In this chapter we will study about a particular types of an algebraic expression, known as polynomials.
Polynomials
An algebraic expression in which the powers of variable are only non-negative integer.
The general form of a polynomial p(x) of degree n as
\[p(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+......+{{a}_{n}}{{x}^{n}}\].
Where \[{{a}_{0}},{{a}_{1}},{{a}_{2}}.....{{a}_{n}}\] are constants with \[{{a}_{n}}\ne 0\] and n is a non-negative integer.
Degree of Polynomials
The highest power of the variable is called degree of the polynomial.
Find the degree of the following polynomials.
(a) \[{{x}^{2}}+7{{x}^{2}}+14{{x}^{6}}+5{{x}^{4}}+5\]
Solution:
The degree of polynomial is 6 because the highest power is 6 in \[14{{x}^{6}}\].
(b) \[{{x}^{2}}y+xy+{{x}^{2}}+x+1\]
Solution:
The degree of polynomial is 3 because the highest power is 3 in \[{{x}^{2}}y\].
(c) \[9{{x}^{3}}+4{{x}^{2}}+4x+5\]
Solution:
The degree of polynomial is 3 because the highest power of variable is 3 in \[9{{x}^{3}}\]. 
Introduction
In the previous classes we have studied that whenever a number is raised to certain power, than we write it in exponential from which tells us that how many times the number appearing in the base is being multiplied by itself and the number of times is being indicated by exponent.
Logarithms are mathematical statement which is used to answer a slightly different question for exponents whose base is a positive real number. This if a is a positive real number other than 1 and \[{{a}^{x}}=n\], then \[x\] is called the logarithm of n to the base a, and the equation may be written as \[x=\log _{a}^{n}\].
Note: If no base is mentioned in a logarithm then it is taken as Logarithms to the base 10 are known as common logarithms. 
Properties of Logarithms
Antilogarithms
The logarithm of a number always contains two parts which are characteristic and mantissa. The integral part is known as characteristic and the decimal part is known as mantissa. Mantissa is always kept positive. The number whose logarithm is \[x\] is called the antilogarithm of \[x\] and is denoted by antilog \[x\].
The value of \[{{\log }_{343}}7\] is:
(a) 0
(b) 7
(c) \[\frac{1}{3}\]
(d) \[\frac{1}{7}\]
(e) None of these
Answer: (c)
Explanation:
let \[{{\log }_{343}}7=x,\] then \[{{343}^{x}}=7\]
\[\Rightarrow \]\[{{({{7}^{3}})}^{x}}=7\]\[\Rightarrow \]\[{{7}^{3x}}=7\]\[\Rightarrow \]\[3x=1\]\[\Rightarrow \]\[x=\frac{1}{3}\]
If \[\log 2+\frac{1}{2}\log x+\frac{1}{2}\log y=\log (x+y)\], then:
(a) \[x=y\]
(b) \[x+y=1\]
(c) \[x=2y\]
(d) \[x-y=1\]
(e) None of these
Answer: (a)
Explanation:
\[\log 2+\frac{1}{2}\log x+\frac{1}{2}\log y=\log (x+y)\]
\[\Rightarrow \]\[\log (2\times \sqrt{x}\times \sqrt{y})=\log (x+y)\]
\[\Rightarrow \]\[{{(x-y)}^{2}}=0\]
\[\Rightarrow \] \[x=y\] 
Introduction
Statistics is the formal science of making effective use of numerical data relating to group of individuals or experiments. It deals with all aspects, including the collection, analysis and interpretation of data, and also the planning of the collection of data, in terms of the design of surveys and experiments.
A statistician is someone who is particularly versed in the ways of thinking necessary for the successful application of statistical analysis. Often such people have gained this experience after starting work in number of fields. This is also a discipline called mathematical statistics, which is concerned with the theoretical basis of the subject.
Data is normally classified into two types: Primary data and Secondary data. The primary data is that data which is collected by the person himself for his own personal use, while secondary data is that data which is collected by others and used by someone else for his or her use. It may be data collected form the books, newspaper, internet or any other sources.
The data may be in the form of raw or grouped. The data which is not arranged in any form is known as the raw data and the data which is arranged in any definite pattern is known as the grouped data. 
Classification of Data
In order to tabulate a large number of data, we use the frequency distribution table. Frequency distribution is of two types:
1. Discrete Frequency Distribution
2. Continuous Frequency Distribution
In discrete frequency distribution method the frequency distribution is carried out with the help of raw data using the tally marks. But in continuous frequency distribution the data is divided into small groups of class interval and corresponding frequency is found. The frequency of a data is defined as the number of times a data is repeated in the given collection of data. The small groups into which the given data is divided is known as its class interval. The difference between the upper limit and lower limit of a class interval is known as its class size.
The class mark is defined as the average of the upper limit and lower limit of a class interval. The cumulative frequency is defined as the sum of all previous frequencies of the class interval.
| Class Interval | Frequency |
| 0 - 5 | 4 |
| 5 - 10 | 10 |
| 10 - 15 | 18 |
| 15 - 20 | 8 |
| 20 - 25 | 6 |
more... 
Cumulative Frequency
The sum of all previous frequencies of a class interval is called as the cumulative frequency of that class interval. The cumulative frequency of the last class interval is the sum of all frequency of the given data.
The Cumulative Frequency Distribution Table for The Data Given below is: 1,1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7,7
Bar Graphs
Bar graph is an excellent way to show results in graphical form that is one time, and isn't continuous - especially samplings such as surveys, inventories, etc. Below is a typical survey asking students about their activities after school. Notice that in this graph each column is labelled - it is also possible to label the category to the left of the bar.
Favorite Activity of Students after School
