Ninth Standard math introduction

Friends, today we all are going to discuss about the overall syllabus of your grade IX mathematics and also going to talk about the most suitable way to solve such type of problems. Sadly, most students start learning mathematics by assuming it as a bundle of recipes to handle certain problems. Eventually mathematical world is a web of facts, concepts, and logical reasoning and, a more efficient approach to understand it is to focus on the principles and connections rather than on isolated facts.

The first approach to understand this important concept are building mathematics in this what students need to do is to reduce their problem to one that he/she have solved before. This principle is used all around in mathematics. The main sayings of this principle is : in order to understand a piece of mathematics student have to understand what preceded it. Moving forward what student need to do is to introduce concepts in a simple context and then generalize them in such a manner that facts and rules that are true in the simple context always remain true in the more general context. Before student go and attempt the solution of a given problem or starts studying a subject, it is worthwhile to think absolutely about what he/she expect their solutions to look like, or what they expect to learn. In general there are two outputs or we can say that two possibilities that are students expectations are met or they are not. In the second situation what students generally do is they make a mistake and now that they are alert to the fact that they can recover from it.

Now I am going to give you an introduction about your IX grade mathematics syllabus. The ninth grade syllabus mostly includes following topics in most generalized manner. The topics to cover are as follows :

1. Algebra
2. Geometry
3. Statistics
4. Probability
5. Pre calculus
6. Calculus
7. Trigonometry

Firstly discuss about the Algebra topic of ninth standard, This topic typically treated in ninth standard and most of the topics are considered as the part of Algebra 1. It includes exponents, radicals, properties of numbers, patterns, number system, evaluation of expressions involving signed numbers, exponents and roots, properties of the real numbers, absolute value and equations and inequalities involving absolute value, scientific notation, unit conversions, solution of equations in one unknown and solution of simultaneous equations, the algebra of polynomials and rational expressions, word problems requiring algebra for their solution, Pythagorean theorem, algebraic proofs, functions and functional notation, solution of quadratic equations via factoring and completing the square, direct and inverse variation, and exponential growth. Some of the few other topics are Gauss-Jordan elimination and kinds of matrix formation that includes Matrix/vector addition and Matrix/vector multiplication.

Now let's have a brief introduction about how to solve a word problem. What student need to do is to look for the statements in the problems that describe equal quantities. Then use of algebraic phrases and equal signs to write equations that make the same statements of equality.

To solve a difficult problem what student need to do is to solve a simpler problem first and then apply whatever formulas or methodologies that student learn to solve the more difficult problem. If student is going to opt this technique then he/she might have to iterate this technique and try to build whole hierarchy of problems to solve a truly difficult problem.

Moving further let's talk about the next topic in the ninth grade mathematics, that is Geometry. Geometry is an important area of mathematics which deals with the shape, size, relative position of figures, and the properties of space. It is all about shapes and their properties. Geometry is of two types : Plane geometry and solid geometry. Plane geometry deals about the shapes on a flat surface like lines, circles and triangles, shapes that can be drawn on a piece of paper whereas Solid geometry is all about three dimensional objects like cubes, prisms and pyramids. In ninth standard Geometry, we all are going to learn the following sub topics that are : Euclidean theories and algorithms, then we study about properties of geometry and a whole concepts of coordinates and transformations. Another important topic which is used in both Algebra and in Geometry that is Pythagorean theorem which helps in calculating the third side in a right angle triangle. Then we are going to understand the basic concept behind the coordinate system and look towards measurement of Geometry like triangle, square etc. In addition to this we will talk about different Geometric concepts and isoceles triangle theorems and rectangular coordinate system. Another important section comes under Geometry is Rate, distance, time, angle measurement and arc length. Inequality theorem along with angles of triangles and polygons along with unit of measures etc. are some of the important topics of Geometry.
Now the next topic of ninth standard mathematics are Statistics and Probability which are interrelated with each other, so I am going to discuss both the topics together. In lots of statistical analysis, results depend on the probability calculation like in many of the census results depend on probability and to represent probability we need statistics help. Probability and Statistics together play an important role in finding out measures of central value, measures of spread, and helps in comparing of two data. These two branches are basically treated as the tools of data analysis. Statistics is the practice or science of collecting and analyzing numerical data in large quantities. It is basically a study of the collection, organization, analysis and interpretation of the data. Probability is a way of telling or expressing knowledge that an event will occur or has occurred. The probability of an event occurring given that another event has already occurred is called a conditional probability. So in this section we deal with Mean, median, mode, range, standard deviation and variance in the numeric sequences. We are going to find the combinations and permutations among the series of numbers along with this we need to represent the data with the help of Data representation methods. In probability section I am going to discuss about types of events, conditional probability etc. Statistical analysis and methods to make inferences and organizing and displaying data along with making predictions from data are the another important topic of ninth grade mathematics.
Now the other two topics related with each other are Pre-Calculus and Calculus. Calculus is basically a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Basically it is the study of 'Rates of Change'. There are two main branches of calculus: Differential Calculus and Integral Calculus. Differential calculus determines the rate of change of a quantity and integral calculus finds the quantity where the rate of change is known. In ninth standard Pre-Calculus section I am going to discuss about the following topics that covers topics like describing functions, representing functions along with Arithmetic and geometric series. The calculus section of mathematics branch includes various mathematical models with their applications to solve problems using multiple approaches along with this it also deals with techniques used to study patterns and analyze data and problems related to personal income and credit. It also comes with problems related to polar and rectangular coordinates along with trigonometric form of complex numbers and mathematical induction. Conic section and fundamental theorem of algebra are also the part of Calculus problem.
The last and final sub topic of ninth grade mathematics is Trigonometry. It is one of the important branch of maths and it also plays an important role in the physics branch. Trigonometry is the branch of mathematics that deals with triangles, circles, oscillations and waves. In mathematical term it is the study of triangles and the relationships between their sides and the angles between these sides. Trigonometric functions are the functions of an angle of a triangle, that are used to relate the angles of a triangle to the lengths of the sides of a triangle. The sub topic of ninth standard trigonometry are Degree and radian measures along with various basic trigonometric functions which I already stated above. Special right triangles means triangles placed in different form or angles and trigonometric values of standard angles are the next important topic of mathematics. Justification of Pythagorean identities are also an important section along with different trigonometric equations is also included in the syllabus.
Whenever student is going through several problems then he should go for work on the easy one first. As this might teach the students something that will surely help them to solve the more difficult problems in a faster and in a better manner. Whenever a student is done with their mathematic question, he or she needs to check the answers as there may be some mistakes in the problem or solution obtained is not an appropriate one. So moving further in next class we again start from the first topic of ninth grade mathematics and try to understand the basic concept behind the Algebra.

Logarithmic Rules and formulas in IX Grade

Mathematics is one of the engrossing subject, still students try to move away from this wonderful subject.  But, my dear friends now, onwards you don’t need to move away from this subject as I am here to help my kids, so that they can easily grasp the concepts of this subject. Now, I will teach you all the math topics. Grade IX is the most important subject of student’s career as it decides where they have to go after this. In this article we will talk about the very crucial topic of algebra taught to students of class IX. Grade IX is the starting class of higher studies and from this class subject become tough and complex. If one has intention then he/she can easily reach to its goal with little efforts. So set your target and start making all possible efforts to reach it. Ok, now, move to the topic that we have to study today, the Logarithmic rules and formulas. Logarithmic is the new topic introduced in Grade IX.

What do you understand by this term what did it mean? No idea, ok , I will tell what is this, Logarithms is an exponent to which constant is raised to obtain a given number. In other words, we can also define it as:
The logarithm of a number represents the quantity as a power to which a fixed number has to be raised to produce a given number. The fixed number is known as base. For example, the logarithm 10000 to base is 4, because 10000 is 10 to the raised power 4 that means:
10000 = 10⁴ = 10 x 10 x 10 x 10.
One more general from we can define it as if x = c^y  then y is the logarithm of x to base b, and is written log_c(x), so log₁₀(10000) = 4.
x = c^y is the same as y = log_bx

In simple words logarithm is just an exponent. Kids don’t get confused, let’s specify it, the logarithm of a number x to base b is just is just the exponent you put onto b to make the result equal x.
Let’s have an example to understand this concept, as you know 6² = 36, is in exponent form (the power is 2) and this is the logarithm of 25 to base 5. In above paragraph I explained you how to write log function, log₆(36) = 2.
Let’s see an example to understand the basic concept of logarithm.
Log₃81 = ?  is the same as 3? = 81,
In the second equation if we calculate the value of unknown than we can easily the value of equation one.
blog b_x = x,

We can read it as the logarithm of x in the base b is the exponent you put on b to determine the value of x as a result.
3x = 81,
x = 81/3,
x = 27,
Now, we can easily put the value of known in the equation 1,
Log₃81 = 27.
In this way we can easily determine the value of unknown in log.

In mathematics we deal with the two types of logarithms: common logarithm and the natural logarithm. The general definition of log is given as: log_ax = N means that a^N = x.
The common logarithm is given as: log_x = log₁₀x. in common log a logarithm is written without a base.

In natural logarithm the logarithm function is written ln and it is given as: ln x = log_ex where, e’s value is approximately 2.718.
For the above statements let x, y, a and b all are positive and a ≠ 1 and b ≠1.

Each language has its well – defined set of rules and if you will not follow the rules properly then you will get wrong result. In the same way mathematics has its set of rules and its topic contains a proper method and rules that you have to follow to get the correct answer. As, in this article we are talking about logarithm so now talk about, Logarithmic rules.

The first rule is known as the inverse property rule that depicts:
log_aa^x = x   and   a(log_ax) = x

Is the product rule that says that: log_a(xy)= log_ax + log_ay

Now, move towards the example so that you can understand what I want to explain you in above theory. Like: expand log₃(2x),
Whenever any problem related to expansion arises it means that you have one log expression with lots of stuff inside it and you have to use the log rules and functions to get the complicated answer. In the above question we have 2x inside the log problem. In this question 2x is in multiplication form so apply the multiplication or product rule to get the answer. When we apply product rule we get:
log₃(2x) = log₃(2) + log₃(x).  And this is the answer of the problem.


The third one is defined as the Quotient rule and according to it:

.

For illustration of the above property let’s take an example and solve it with this you came to know how to use division or quotient rule.

Simplify: log₄(16/x).

In this you can easily find that we have to apply division rule as the logarithmic function contains (16/x) in division form.
According to division rule we get:
log₄(16/x) =  log₄(16) - log₄(x),
The first term on the right hand side can be simplified to exact form or value, by applying the basic rule of the log that we discussed in the definition of the log.
log₄(16) = 2, so, the final answer is:
log₄(16/x) = 2 - log₄(x).
After you get the answer don’t forget to cross check it and see that you properly expand all the terms or not.

The fourth rule is known as the power rule that shows:
log_a(xp) = p log_ax.
To understand this property properly let’s take an example and solve it. log₅(x³)
The exponent inside the log can be taken out as a multiplier
log₅(x³) = 3.log₅(x) = 3log₅(x)
This is the simple property and according to this rule we take the power in the front side of log as a multiplier. This is simple one but most commonly used property of log used in almost each and every problem.


The last one is defined as the change of base formula:

Example for this: log₃[4(x - 5)² / x⁴ (x -1)³],
In this we will solve the question using the above property. In the first step we proceed as:
log₃[4(x - 5)² / x4 (x -1)₃] = log₃[4(x- 5)₂] – _log3[x4(x -1)³],
                                         =[ log₃(4) + log₃[(x -5)²)] - log₃(x⁴) + log₃[(x - 1)³)]
                                         = log₃(4) + log₃[[(x - 5)²] - log₃(x⁴) – log₃ [(x - 1)³]
                                      = log₃(4) + 2 log₃[ (x -5)] - 4log₃ (x) - 3log₃[(x - 1).
Thus, this is the final answer of the above problem.

While solving logarithmic problem you all must keep few things in mind so that you can solve the question properly without any problems without mistakes.
log_a (x + y)   ≠ log_ax + log_ay,
log_a(x - y)   ≠ log_ax – log_ay.
Now, let’s see one more example that students can understand the different type of problem that occurs in logarithms and as you know example is the best way to learn and understand different concepts rule or functions.  So here is one more example for you and see the step involved while solving the problem and with this you can easily practice lots of questions,
log₂ (8x⁴ / 5)
Solution:
Step 1: 5 is divided into the 8x⁴ , so first split the numerator and denominator by using subtraction, on doing this we get:
log₂(8x⁴ / 5) = log₂ (8x+) - log₂(5),
Step 2: in the next step don’t take the exponents out; it is only on the x, not on the 8 and we can take only those exponents that are present on every term inside the log. So in this split the factors using the addition,
log₂ (8x⁴ ) - log₂(5)= log₂(8) + log₂(x⁴) - log₂(5),
Step 3: the x has an exponent so take it in front of log as a multiplier,
log₂(8) + 4log₂(x) - log₂(5),
Since the power of 2 is 8, I can simplify the first log to an exact value:
log₂(8 ) + 4log₂(x) - log₂(5),
3 + 4 log₂ (x ) - log₂(5)
Now, each log contains single value so the expression is simplified and the answer of the problem is, 3 + 4 log₂(x ) - log₂(5).
This is all about logarithmic rule and logarithm formula and for detail switch to online help.

System of Equations and Inequalities of Ninth Grade

Friends today we all are going to understand about the basic concept behind one of the most interesting and a bit complex topic of Ninth Grade mathematics that is system of equations. Till eigth grade we had equations but they were easier to solve but as we are moving towards the ninth grade, equations become little complex. We all now have the basic knowledge of equations as used in mathematics, but let us do a quick recap. In simple mathematical manner we can say that an equation consist of two algebraic expressions and 'is-equals-to' symbol (=) is used between the two expressions. An algebraic expression is one which consists of variables and constants and if the equation is true for all the variables used then it is called as identity. For example

X + Y = Y + X (an example of Identity)

Now the next question that comes to our mind is, How to solve an equation? Lets take an example to understand it better: an equation 4x + 7 = 31. now we have to figure out the value of variable x where this equation holds true and the procedure of figuring out this unique value of x is what we called as solving an equation.

4X + 7 = 31 can be written as 4x = 31 – 7 = 24

now x = 24/4 = 6.

We already know that a linear equation is the one in which we always get a straight line while graphing it. Also the linear equation in one variable is an equation of the form :

ax = b where a and b are the constants and x is the variable whose value need to be determined.

Now I am going to tell you about system of equation of grade nine or the ninth grade that is the next step after the linear equations. but before we move on we should understand what is a system of equations: A systems of equations come with the relationship between two or more functions, which can be used to form a number of real-world situations. It is basically a collection of two or more equations with a same set of unknowns. An equation can be linear or Non linear. In the ninth grade we have to learn about system of linear equations. A system of Linear equations is a collection of linear equations that come with the same set of variables.

An example will help us in understanding it better, suppose there are two variables “a” and “b” and we have two options.

a + b = 5

2a – b = 1

Lets starts it by finding out whether both the equations are satisfied if a = 2 and b = 3. This is easy to check as I know how I can get the result for the equations, but how we are going to arrive the answer if nobody gives us the solution? So, the real issue is how to find out the exact solution for a system of equations.

In general our course books provide us two methods of solving a linear equation that are sibstitution and elimination method (also called as Gaussian Elimination). Substitution method works by solving on equation for one variable and substituting its value in other equations. This method is basically used for small systems and it can be applied in nonlinear systems to get the desired results. Elimination or the Gaussian Elimination method becomes increasingly important while solving large linear system of equations.
An example will definitely help in a better understanding, lets continue with the same example. Multiply the first equation with 2 in both the sides. Now the equation becomes 2a + 2b = 10 and the another equation becomes standard or unchanged as : the two equations are 2a + 2b = 10
2a – b = 1
As in the above equations the variable “a” has the same coefficient in both equations. Now the next step is to subtract 10 from both the sides of second equation. It looks like
2a – b – 10 = 1 – 10
Now, as we already know that 2a+2b = 10 so, lets substitute number 10 with this value :
2a – b - (2a + 2b) = -9.
Now we need to simplify the above equation in following manner :
2a – b – 2a – 2b = -9
So we get – b – 2b = -9
And further simplifying it we get : - 3b = -9
Divide -3 from both the sides of the equation, we get
b = 3
Now substitute the value of b in the first equation to get the value of another variable :
2a + 2(3) = 10
2a + 6 = 10
2a = 4
a = 2.
In the end we can check our result by substituting the values of “a” and “b” in the equations. Clearly substituting a = 2 and b = 3 in the equations if the values present true answer for the equation then the result is correct (means LHS should be equivalent to the RHS).

Linear systems can be represented in matrix form as the matrix equation: Ax = b. Here A is the matrix of coefficients, x is the column vector of variables, and b is the column vector of solutions. An example to show a system of equations:

2x + 3y – z = 1

3x – 4y – 7z = 8

-x + ¾ y – 2z = 4

It is a system of three equations in the three variables x, y, z.

Next lets think of a system of equations that has more than two equations, say “n” equations, to solve such kind of problem we need to pick one of these equations, it could be the first one or any in the system of equations and a particular variable. Subtract suitable multiples of our selected equation from the other equations so that the selected variable no longer remains in the resulting n – 1 equations with n – 1 unknowns. Now in next step we need to repeat the same process until we get a single equation in a single unknown (means we need to further simplify the equation to get the simplest form of single equation). This whole process is known as Gaussian Elimination.

The next important topic to be discussed is Inequality. In 8^th and 9^th standard inequality problems are very easy to solve. As in ninth grade the syllabus has several problems involving linear inequalities. Before we go on to solving these problems, we should understand the inequality. An inequality tells that two values are not equal. For example a ≠ b shows that a is not equal to b. Slope formula plays an important role in graphing linear inequalities. So we need to know is this slope formula means and how to use it. Slope of a line describes the steepness, incline or grade of the straight line.

The slope through the points (x1, y1) and (x2, y2) is given as Slope formula.

( m = y2-y1/x2-x1 ) where x1 is not equal to x2.

In general slope intercept form denotes the formula : y = mx + b.

Where m = slope of the line

b = y intercept.

If we what to understand it in detail lets start with basic information that an inequality consists of two algebraic expressions combined by one of the inequality symbols like < means less than and the symbol > means greater than and the symbol ≦ or ≤ less than or equal to etc. There symbol ≠ is also used, but the other cases occur more frequently.

A linear inequality describes an area of the coordinate plane that has a boundary line. In simple words a linear inequality represents everything on one side of a line on a graph.

In mathematics a linear inequality is an inequality which involves a linear function

for better understanding here is an example -4 < 1

If we multiply -1 from both the sides we get the values 4 and -1, but

4 > -1.

The multiplication with negative value reverses the inequality sign. So generally when we solve problems problems involving inequalities the procedure is very similar to the way we the equality problems. But one thing that we have to always be careful about is to reverse an inequality sign when we multiply on both sides with the same negative value or factor.

Here is one more ninth grade example to understand it better :

3a + 4 < 5

To solve such type of inequality we need to subtract 4 in both the side :

we get : 3a < 1

now further simplifying it we need to subtract both the sides by 3

a < 1/3 is the required solution.

one more example to understand the basic concept behind reversing the symbols. This example shows you that how to reverse a symbol or sign when negative value is used in the equation.

Equation is : 3a + 4 < 5a – 2

now what we have to do is to subtract 4 and 5a from both the sides

we get – 2a < -6

then in next step we simply needs to divide both the sides by - 2 and just reverse the sign to obtain :

we get a > 3

But once you get the solution always check it, for this  we just need to replace the inequality sign with an equality sign and just substitute the value of our final result in the equation. If the result represent true value for the equation then your result is right else you will have to do it once again.

Polynomial Functions in Grade IX

Friends today I am going to discuss about one of the most interesting and a bit complex topic of Grade IX math that is Polynomial. Before proceeding further let’s talk about Polynomials. In earlier standards what I learn that a Polynomial is basically a term which deals in almost every type of mathematical equations or statements. The most common terminologies used in polynomial expressions are monomials, binomial and trinomials. Algebraic equations with all variables having whole number, exponents or powers are called polynomials. The expressions in which the power of variables are negative and rational numbers are not polynomials. Algebraic expression having single term known as Monomial and expression with two terms are known as Binomial whereas expressions having three terms are known as Trinomials.
Let’s see some of the examples of the polynomial equations to understand it better.
10xyz: Monomial
3x + 7y: Binomial
3x + 7y – c: Trinomial

Now considering 9th grade, I am going in deep with polynomials. In simple mathematical manner, a polynomial “p” (in any of the variable x) is an expression or a function that can assess by associating the variable and by any means by a finite number of additions, subtractions and multiplications. The most important note is that the list excludes division(for example, any number like ¾ would be considered as a constant). It also ignores radicals or values come under square root (√) (for example √5 ) considered as a constant. Most of the times we use the words or we pronounced like polynomial expression or polynomial function, but mostly we use the word Polynomial.
A polynomial can be written in the standard form as:
$displaystyle p(x) =a_nx>n + a_n-1x>n-1 + ldots + a_1 x + a_0qquad(*) $

Here a_i are constants and also called as coefficients of the polynomial. Integer n >= 0 is called the degree of the polynomial. In most of the time this context can be stated 

 that and we can adopt that convention for the purposes of this class. The form (*) shows the origin of the word polynomial which is Greek for many terms.

Any function like p(x) = 0 is also a polynomial. It is also known as zero polynomial (or we can also state it as the zero function). The degree of the function is undefined (-1 or negative infinity). Few of the examples which illustrate the above concepts:
1: P(y) = y² – 3 is a polynomial of degree n = 2 and it is written in the standard form that is a₂ = 1, a₁ = 0 and a₀ = -3.
2 : p(y)= by⁵ + 2y⁴ – 3y² + 4y - √2 is a polynomial of degree 5 with a₅ = b, a₄ = 2, a₃ = 0, a₂ = -3, a₁ = 4 and a₀ = - √2.
Polynomial functions comes equipped with terms, factors, variables, and constants. Let us explore about all these required objects to form a polynomial function, When numbers are implemented with addition or subtraction than they are said to be terms and terms are of two types Like terms or Unlike terms. Term that has the same power of the same variables is called Like term. The terms used in an expression that do not contain the same power of the same variables are called unlike terms. In an expression if the product of the numbers are used then the expression is called as factors. Variables are just representing a symbol which use different values under it whereas constant is a single value symbol.

Now I am going a bit deep with Polynomials as I am going to discuss about, how to combine polynomials. Students can add, subtract and multiply polynomials to get a new polynomial. Another point to understand is that the ratio of any two polynomials is customarily not a polynomial. Sometimes students try to memorize a number of rules for the various operations. Instead of recognition of a number of rules, student should recognize that these are just the ordinary rules underlying all algebra, such as the associative laws, distributive laws and the commutative laws of multiplication and addition. Let’s take an example to illustrate it well.
P(y) = y² + 1 and q(y) = y – 1.
The new function t can be defined as follows (addition, subtraction and multiplication)
T = p + q where T(y) = y² + 1 + y – 1 = y² + y
T = p – q where T(y) = y² + 1 – (y – 1) = y² – y + 2
T = p x q where T(y) = (y₂ + 1)(y – 1) = y³ – y² + y - 1
One thing to remember is that students multiply powers with the same base by adding the exponents. This shows that the degree of the product of two polynomials is the sum of the individual degrees. If student adds (or subtracts) two polynomials of different degrees then the degree of the sum (or difference) is the larger of the two individual degrees. If the two polynomials have the same degrees then the degree of the sum or difference is that same degree unless the leading coefficients cancel, in which case the degree of the sum or difference is less. Student should not require to memorize these facts or rules as they need to practice a lot and make up some new examples and then think about the mechanism of the operation, and at last students need to work out the details from the understanding when student need them.

Polynomials occur all over the map in applications. Various techniques have been developed which helps us to work with polynomials and tells how to exploit their special architecture.

Now we all are going to see the basic fundamentals behind evaluating a polynomial. It can be understood more properly while taking a direct problem and try to solve the problem. Let’s take an example : p(y) = y³ – 6y² + 11y – 6
To solve this equation if we put the value of y as 1, then the result for the polynomial is true.
P(1) = 1 – 6 + 11 – 6
P(1) = true for the polynomial.
Suppose if student wants to know what is the result for P(7). Student can evaluate P(7) in the above polynomial, multiply with the coefficients, and add the individual terms to get.
P(7) = 7³ – 6 x 7² + 11 x 7 – 6
= 343 – 6 x 49 + 11 x 7 – 6
= 343 – 294 + 77 – 6
= 120.
The above procedure looks a bit clumsy. The solution or the evaluation of the above polynomial becomes much simpler if students rewrite the above polynomial p in much simpler way, as
p(y) = ((y – 6)y + 11)y – 6
Evaluating p is much easier in the above form, as students don't have to compute powers. If we do the same operation with the new form polynomial then we can get the solution faster and in better manner.
P(7) = ((7 – 6)7 + 11)7 - 6
= 18 x 7 – 6
= 120.
In the second way of solving polynomial, students get the same answer but the main difference is computation. Here the computation is simpler and it also involves some basic operations. The most important advantages of this operation or form are more pronounced or form for the polynomials of high degree. The above technique to solve polynomials work in general and it is described in the literature as synthetic division, nested multiplication or Horner's Scheme. It's a good rule of thumb that if something is known under several names it is usually powerful or otherwise important. This will be explained in another class.

Quadratic equations are also known as polynomial equations of degree two. A polynomial equation of the second order is known as Quadratic equation. The general form of quadratic equation is : ax² + bx + c = 0
where x is a variable and a, b and c are constants. Here a is quadratic coefficient, b is a linear coefficient and c is a constant term or we can say that it is a free term. Quadratic equations can be solved by using following methods: factoring, completing the squares, graphing, Newton's method, and with the help of Quadratic formula.

The Quadratic formula. Quadratic equation is ax² + bx + c = 0 and it has the solutions
here the expression under the square root sign is known as discriminant of the quadratic equation. Discriminant is denoted by the upper case Greek delta.
Delta = squared b – 4ac .
If the discriminant is zero then there is only one exact real root, also known as double root.
X = -b/2a.
The ‘±’ symbol indicate as ‘plus or minus’, which means that we need to work out the formula twice, once with a plus sign in that position, then again with a minus sign.
x=frac-b pm sqrt b>2-4ac2a,