Polynomials

An expression with a single term is called as monomial, with two terms as binomial and with three terms as trinomial. If the number of terms is more, then such expressions are given a general name as polynomials, the word ‘poly’ means ‘many’. So it can also be written as poly-nomial to emphasize the meaning. Therefore, a poly-nomial is an expression that contains a number of terms consisting variables with constant coefficients. The terms of expression are usually arranged in descending order of the variable powers. Since a constant can also be expressed with the variable power as 0, a constant term can also be a part of a poly-nomial.
But as per convention in algebra, the definition of a polynomial includes certain restrictions. A polynomial can be built up with variables using all operations except division. For example, x³ – (2x + 3)/(x) + 7 cannot be called as a polynomial. However, this restriction applies only division by a variable and not for division by any constant, because such divisions can be considered as equivalent to fractional coefficients. The other restriction is that the exponents of the variables can only be non-negative integers.

A polynomial is generally an expression but acts as part both in equations and functions and such equations and functions are named with prefix ‘poly’ in general. In fact we convert a polynomial function to an equation while attempting to find its zeroes. Therefore, it is imperative to know how to factor a polynomial so that the zero product property can be used to determine the solutions. Using the zero product way is the easiest way to find the solutions of the variables.

Polynomial equations with a single variable with degrees 1 and 2 can easily be solved and the latter type is more popularly known as ‘quadratic’ equation. Mostly quadratic equations are possible to solve by factoring but even otherwise it can be done by using the quadratic formula. But equations with higher degrees are not all that easy to solve. But thanks to the great work by the mathematicians, there are ways to do that. Let us see some of the helpful concepts enunciated by the mathematicians.

As per fundamental theorem of algebra, the number of roots (the number of solutions of variables when equated to 0) of a polynomial is same as the degree of the same. This concept guides us to do the complete solution. We must also be aware that in some cases the solutions or some of the solutions may be imaginary. To get an idea on this, Descartes’s rule of sign changes helps. As per this rules we can figure out the number of real solutions, both positive and negative. Subsequently we can figure out the imaginary solutions with the help of fundamental theorem of algebra.

The rational zero theorem helps us to know what are the possible zeroes of a function. By a few trials, we can figure out a few zeroes and can reduce the polynomial to the level of a quadratic. Thereafter, the remaining zeroes can easily be figured out.

In addition, these days there are many websites advertising as ‘Factor Completely Calculator’ to help us in finding the solutions of a polynomial.

Correlations

Zero Order Correlation :- Zero order correlation means there is no correlation between the two quantities. They vary independently.  If classes in one variable are associated by the classes in the other, then the variables are called correlated. The correlations is said to be perfect if the ratio of two variable deviations is constant. Numerical measure of correlation is called co- efficient of correlations. A group of n individuals may be arranged in the order of merit with respect to some characteristics.

The same group would give different order for different characteristics. Consider order corresponding to two characteristics A and B. The correlations between these n pairs of ranks is called rank correlation in characteristics A and B for the group of individuals.  When the variation of the value of one variable becomes the cause of the variation of the value of other variable then it is called the correlation between the two variables. When the variation between two quantities is directly proportional then it is called positive correlations. It means when one variable increases other also increases or when one variable decreases other variable also decreases.

When the variation between two quantities is indirectly proportional then it is called negative correlations. It means when one variable increases other decreases or when one variable decreases other variable increases. For example the production of any quantity is indirectly proportional to the cost of that quantity. When the temperature increases the charge of the electricity bill also increases. This is the example of positive correlations.

On the other hand during the summer season the length of night hours decreases. It means the day hours are increasing and night hours are decreasing. During the cold season the day hours are decreasing whereas the night hours are increasing. There is a negative correlation between the day and the night hours. The reception of the radio signals is indirectly proportional to the distance from the transmitter. The reception near the transmitter is better than the reception at a distance. As the distance between the transmitter and the receiver increases the signal strength in the receiver decreases. It means the signal strength is indirectly proportional to the distance from the transmitter.

This is an example of the negative correlation. Zero order correlation means there is no correlation and all the quantities have their own graph.

Correlation Equation: - Correlation equations are written to find the order of correlations. We can find the perfect positive correlation, negative correlation or zero order correlations.  The numerical measure of correlation is called co- efficient of correlation and is defined as
r = (∑XY)/(n σx σy ) =(∑XY)/√(∑X² ∑Y²), where X and Y are the deviation from the mean positions.

Deviations are used to find the lower and upper bound of that quantity. Let a resistance value is written as 200 ± 5%.

The lower value of the resistance is equal to 190 and upper value is 210 ohm.
σ²x=(∑x²)/n  ,   σ²y=(∑y²)/n

Where, X = deviation from mean, ¯x=x-¯x

Y = deviation from mean, ¯y=y-¯y
σ_(x =  Standard deviation of x series )
σ_(y =  Standard deviation of y series )
n = number of values in two variables

We can also use the direct Method by substituting the value of σ_(x ) and  σ_(y ) in the above mentioned formula.

r = (∑XY)/(n σx σy ) = (∑XY)/√(∑X² ∑Y²)          or (n∑xy- ∑x∑y)/√(n∑x²-(∑x )²4 ×{n∑y²-∑x²}

Where r is the coefficient of correlation which can be used to find the rank of two quantities.

Vectors

In this section we are going to understand the basic concept of vectors and we will see a very important operation that can be done on vectors which is cross product. 

Let us see what are vectors first.Vectors are used to represent quantities where magnitude and direction both are essential to define a quantity properly. A vector hence represents magnitude and direction of a quantity. For example quantities like velocity and acceleration needs to be defined with magnitude and direction as well. A velocity of 5 km/hr due east represents that the object is moving with a speed of 5 km/hr in east direction. This means that velocity is a vector quantity. 

A vector is denoted graphically by an arrow where the head of the arrow represents the direction of the vector and its length represents the magnitude of the vector. The length of a vector is used for comparing magnitude of 2 or more vectors. A vector with shorter length will have smaller magnitude than other vectors. 

A vector a can be represented in 3 dimensional space as:
 a= a₁i+a₂j+a₃k

Here i, j and k are unit vectors in the x, y and z direction respectively. 

Let us see how cross product can be performed on the vector  Cross product is also known as vector product. This is due to the fact that the result after the product is also a vector. It is a binary operation on 2 vectors in 3 D space. The resulting vector is perpendicular to both the vectors and thus it is perpendicular to the plane in which initial vectors lie. 

If the direction of the two vectors is same or they have zero magnitude or length, then the cross product of the two vectors results zero. The magnitude of the cross product vector is equal to the area of the parallelogram which has the two vectors as its sides. If the two vectors are perpendicular and form a rectangle, then the resulting magnitude will be the area of rectangle i.e. product of lengths of the vectors. 

The operation of cross product is denoted by ×. If the two vectors are ‘a’ and ‘b’ then the cross product of the two vectors is denoted as a×b. Note that a×b is not equal to b×a. The resulting vector of this cross product denoted by c is a vector that is perpendicular to both the vectors. Its direction can be given by using right hand rule. Formula for Cross Product of Two Vectors is given as:
a×b=|a||b|sinӨ n

Here Ө is the smaller angle between the vectors a and b. |a| and |b| denotes the magnitude of the vectors a and b respectively. n here represents the unit vector perpendicular to the plane of a and b. Its direction can be given by right hand rule. The right hand rule says that if a×b is the cross product that you are calculating then role your fingers of right hand from vector a to vector b, the direction of your thumb will give the direction of resulting vector as shown below:

If the vectors a and b are parallel to each other, then the cross product of then is a zero vector as sin0 = 0
Cross product is considered as anti commutative. This means that a×b = -(b×a) 

Algebra Word Problems

In the field of mathematics where many forms are present algebra is one such form that deals with the basic concepts of Mathematics. We all know that Maths is a game of mathematical operations, so to deal with addition, subtraction, multiplication, division with the known and unknown values of the variables is known as algebra. In a simple way to say is that when we play computer games then we deal with jumping, running or finding secret things and in Maths we play with letters, numbers and symbols to extract the secret things. Now let’s see what is included in algebra.

Get easy steps to solve Algebra Problems (click here)


WHAT IS THERE IN ALGEBRA?
1. Algebraic terms – Algebraic terms are those terms which are infused with the values of variables. For example 3a.
In this example 3 is the numerical coefficient and a is the variable. The main work of numerical coefficient is also to deal with + 3 or – 3 of the variable.

2. Algebraic expressions – Algebraic expressions deal with a logical collection of numbers, variables, positive or negative numbers in the mathematical operations. For example we can write an expression – 3a + 2b.

3. Algebraic equations – Algebraic equations refer to the equivalency of expressions considering both the left hand side and the right hand side. For example an equation is x – 35 = 56k2 + 3. Over here both left hand side and right hand side are dealing with expression. If both the expressions are equal then that is said to be an equation.

ALGEBRA WORD PROBLEMS

The main part in Algebra is to solve the word problems, which we also did in Pre Algebra Word Problems.  It can be harder if you don’t think and do it but it can be easy if you think and do it. If we say in English how are you and the simple answer to the question is I am fine which can be simply understood but to convert the English language in to the language of maths requires certain techniques. You need to practice a lot with proper understanding of the translations. Let’s go step by step.

1. First of all read the entire word problem carefully. Make it a point that we should not start solving the problem by just reading half of a sentence. Read it properly so that you can gather the information that what you have and what you still need to have.

2.  Then, pick the information so that it can be changed into certain variables. Some might be known and some might be unknown. If required then draw pictures and symbols in the rough work so that you can deeply understand the problem without leaving any loopholes.

3. Note the key words as for different mathematical operations there are different key words –

a) Addition – increased by, total of , sum, more than, plus
b) Subtraction – decreased by, minus, less than, fewer, difference between
c) Multiplication – of, times. Multiplied by, product of, increased or decreased by the factor of
d) Division – per, out of, quotient
e) Equals -  is, gives, yields, will be


4. After going though these expressions carefully you can solve various word problems logically. Limit Problems are part of Calculus.

Rational Inequalities

Rational Inequalities : - The rational expressions are written in the P(x)/q(x) form where p(x) and q (x) are the polynomials i.e p(x), q(x)∈z(x), and q(x)≠0 are known as rational functions. the set of rational function is denoted by Q (x).

Therefore Q(x) = (p(x))/(q(x)) (where p(x), q(x) ∈z(x) and q(x) ≠0. for, if any a(x)∈z(x), then we can write a(x) as  ( a(x))/1, so that a(x)∈z(x)  and therefore,(a(x))/1  ∈Q(x);i.e  a(x)∈Q(x). The rational function example is given below.   

(2x+4)/(x²+5x+8)>0
(ax+b)/(cx+d)<0 p="">
Rational inequalities means the left hand side is not equal to the right hand side of the equation. The inequalities are associated with the linear programming.  The step by step procedure to solve the inequalities is given here. 

The very first step to solve the inequality problems is to write the equation in the correct form. There are two sides in the equation, left hand side and the right hand side. All the variables are written in the left hand side and zero on the right hand side. The second step is to find the critical values or key. To find this we have to keep the denominator and the numerator equal to zero.

The sign analysis chart is prepared by using the critical values. In this step the number line is divided into the sections. The sign analysis is carried out by assuming left hand side values of x and plugs them in the equation and marks the signs. Now using the sign analysis chart find the section which satisfies the inequality equations.

To write the answer the interval notation is used. Let us solve an example (x + 2) / (x² – 9) < 0. Now put the numerator equal to zero, we get x = -2. By putting the denominator equal to zero we get x = +3 and x = -3.  Mark these points on the number line. We get minus three, minus two and plus three on the number line.
Now try minus four, plug the value of x equal to minus four we get – 0.286.

Plug x equal to minus 2.5 we get + 0.1818. Now plug 2 in the equation we get minus zero point eight.
Now we try x equal to plus four which is at the right hand side of the number line. We get plus 0.86. The sign is minus, plus and plus sign. The equation is less than zero. To get less than zero we need all the negative values of the equation.

For x equals to plus three and minus three the equation is undefined. For x equals to minus two the equation is zero. The equation is satisfied by the value of x which is less than minus three. Therefore the answer which is the solution of the equation is minus infinite to minus three.

Rational Expressions Applications : - The equations which do not have equal sign are called the expression. The expressions are used to simplify the equations. The expressions are written to form the conditions of the problems.

According to the direction of the question the expressions are formed to solve a problem.