Perimeter: Length of boundary of a simple closed figure.
Perimeter of Rectangle = 2(l +b) Perimeter of Square = 4a Perimeter of Parallelogram = 2(sum of two adjacent sides) Area: The measure of region enclosed in a simple closed figure.
Area of a trapezium = half of the sum of the lengths of parallel sides × perpendiculardistance between them.
Area of a rhombus = half the product of its diagonals. (Scroll down to continue …)
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A quadrilateral has 10 parts – 4 sides, 4 angles and 2 diagonals. Five measurements can determine a quadrilateral uniquely. (Scroll down to continue …)
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Curve: Curve is a figure formed on a plane surface by joining a number of non linear points without lifting a pencil.
Open Curve: An open curve is a curve which does not end at the same starting point or which does not intersect itself.
Closed Curve: Closed curve is a curve which intersects itself or which starts and ends at the same point.
Simple Closed Curve: A simple closed curve is a closed curve that does not intersect itself.
Polygon: Polygon is a closed figure bounded by three or more line segments such that each line segment intersects the other two line segements at exactly two other points (vertices) as shown in the following figures.
Polygons are classified into two types on the basis of interior angles: as (i) Convex polygon (ii) Concave polygon
(a) Convex Polygon: In this case, each angle is either acute or obtuse (angle < 180 o) as shown in the following figures.
Concave Polygon: In this case, any one angle is reflex (angle > 180°) and one diagonal is outside the polygon as shown in the following figures.
On the basis of sides, there are two types of polygons :
(a) Regular Polygon: A convex polygon is called a regular polygon, if all its sides and angles are equal as shown in the following figures.
Each angle of a regular polygon of n-sides =
Important results on polygon :
For a regular polygon of n sides (n > 2).
(b) Irregular Polygon: A polygon in which all the sides are unequal as shown in the following figures,
Triangle :
A simple closed figure bounded by three line segments is called a triangle, it has three sides as AB, BC and AC; three vertices as A, B and C and three interior angles A, ZB and ZCand the sum of all angles is 180°.
i.e., <A + <B + <C = 180°
Quadrilateral:
A simple closed figure bounded by four line segments is called a quadrilateral, it has four sides i.e.,
AB, BC, CD and AD and four vertices as A, B, Can d D and the sum of all angles of a quadrilateral is 360.
Practical Geometry
Five measurements can determine a quadrilateral uniquely.
A quadrilateral can be constructed uniquely if the lengths of its four sides and a diagonal is given.
A quadrilateral can be constructed uniquely if its two diagonals and three sides are
known.
A quadrilateral can be constructed uniquely if its two adjacent sides and three angles
are known.
A quadrilateral can be constructed uniquely if its three sides and two included angles
are given.
Quadrilateral
Quadrilateral is a closed figure with four sides.
Characteristics of a quadrilateral
Angle Sum Property of a Quadrilateral:
Qudrilateral is a four sided closed figure.
Sum of all angles of a quadrilateral is 360°.
Types Of Quadrilaterals
Classification of quadrilaterals
Quadrilaterals are broadly classified into three categories as:
(i) Kite
(ii) Trapezium
(ii) Parallelogram
Kite:
(i) Kite has no parallel sides
(ii) Kite has a pair of equal adjacent sides.
(ii) It is not a parallelogram
Characteristics Of Kite:
Perimeter Of Square
Area Of Kite
Trapezium:
Trapezium is a quadrilateral with the following characteristics:
(i) One pair of opposite sides is parallel to each other.
(ii) The other pair of opposite sides may not be parallel to each other.
Characteristics Of Trapezium
(i) Sum of all angles of a quadrilateral is 360°.
(ii) One pair of opposite sides is parallel to each other.
(iii) The other pair of opposite sides need not be parallel to each other.
Types Of Trapezium:
Quadrilaterals are broadly classified into two categories as:
(i) Isosceles Trapezium.
(ii) Scalene Trapezium.
(i) Right Trapezium.
Isosceles Trapezium:
Isosceles Trapezium is a quadrilateral with the following characteristics:
(i) One pair of opposite sides is parallel to each other.
(ii) The other pair of opposite sides are equal.
(iii) The other pair of opposite sides need not be parallel to each other.
Isosceles Trapezium is a trapezium with the following characteristics:
(i) One pair of opposite sides is parallel to each other.
(ii) The other pair of opposite sides are equal.
(iii) The other pair of opposite sides need not be parallel to each other.
Characteristics Of Isosceles Trapezium
(i) Sum of all angles of a quadrilateral is 360°.
(ii) One pair of opposite sides is parallel to each other.
(iii) The other pair of opposite sides are equal.
(iv) The other pair of opposite sides need not be parallel to each other.
Scalene Trapezium:
Scalene trapezium: Classified by the length of the legs or the measurement of their angles.
Characteristics Of Scalene Trapezium
Right Trapezium:
Right trapezium: Has one pair of parallel sides and one pair of right angles.
Characteristics Of Right Trapezium
Perimeter Of Trapezium
Area Of Trapezium
Parallelogram:
Parallelogram is a quadrilateral with the following characteristics:
(i) Two pairs of opposite sides are parallel to each other.
(ii) Two pairs of opposite sides are equal in length.
Characteristics of a parallelogram
(i) Sum of all angles of a Parallelogram is 360°.
(ii) Two pairs of opposite sides are parallel to each other.
(ii) Two pairs of opposite sides are equal in length.
(ii) Two pairs of opposite angles are equal.
(iii) Diagonals bisect each other.
(iv) Diagonals need not be equal to each other.
(v) Diagonals divide it into two congruent triangles.
Types Of Parallelogram
Parallelograms are broadly classified into three categories as:
(i) Rectangle
(ii) Rhombus
(iii) Square
Perimeter Of Parallelogram
Area Of Parallelogram
Rectangle:
Rectangleis a quadrilateral with the following characteristics:
(i) Two pairs of opposite sides are parallel to each other.
(ii) Two pairs of opposite sides are equal in length.
(iii) All four angles are right angles. (each angle is 90 o).
Characteristics Of Rectangle
(i) Sum of all angles of a quadrilateral is 360°.
(ii) Two pairs of opposite sides are parallel to each other.
(ii) Two pairs of opposite sides are equal in length.
(iii) All four angles are right angles. (each angle is 90 o).
(iii) Diagonals bisect each other.
(iv) Diagonals are equal to each other.
(v) Diagonals of a rectangle divide it into two congruent triangles.
Conclusions:
Every Rectangle is a Parallelogram. But Every Parallelogram need not to be a Rectangle.
Condition for a rhombus to be a square:
If all four angles of a parallelogram are right angles. (each angle is 90 o), the parallelogram becomes a Rectangle.
Perimeter Of Rectangle
Area Of Recatangle
Rhombus:
Rhombus is a quadrilateral with the following characteristics:
(i) Two pairs of opposite sides are parallel to each other.
(ii) All four sides are equal in length.
Characteristics Of Rhombus
(i) Sum of all angles of a quadrilateral is 360°.
(ii) Two pairs of opposite sides are parallel to each other.
(ii) All four sides are equal in length.
(ii) Two pairs of opposite angles are equal.
(iii) Diagonals bisect each other.
(iv) Diagonals need not be equal to each other.
(v) Diagonals divide a Rhombus into two congruent triangles.
Conclusions:
Every Rhombus is a Parallelogram. But Every Parallelogram need not to be a Rhombus.
Condition for a rhombus to be a square:
If all the sides of a parallelogram are equal, the parallelogram becomes a Rhombus.
Perimeter Of Rhombus
Area Of Rhombus
Square:
Square is a quadrilateral with the following characteristics:
(i) Two pairs of opposite sides are parallel to each other.
(ii) All four sides are equal in length.
(iii) All four angles are right angles. (each angle is 90 o).
Characteristics Of Square
(i) Sum of all angles of a quadrilateral is 360°.
(ii) Two pairs of opposite sides are parallel to each other.
(iii) All four sides are equal in length.
(iv) All four angles are right angles. (each angle is 90 o).
(v) Diagonals bisect each other.
(vi) Diagonals need not be equal to each other.
(vii) Diagonals divide a Rhombus into two congruent triangles.
Conclusions:
Every square is a Rhombus. But Every Rhombus need not to be a square.
Condition for a rhombus to be a square:
If all the angles of a rhombus are right angles (euqal to 90o), the rhombus becomes a square.
2. Every Square is a prallelogram. But Every prallelogram need not to be a square.
Condition for a prallelogramto be a square:
(i) If all the angles of a parallelogram are right angles (euqal to 90o), and all the sides of a parallelogram are equal in length, the parallelogram becomes a square.
3. Every Square is a rectangle. But Every Rectangle need not to be a square.
Condition for a Rectangleto be a square:
If all the sides of a Rectangle are equal in length, the Rectangle becomes a square.
If all the sides of a parallelogram are equal, the parallelogram becomes a Rhombus.
Perimeter Of Square
Area Of Square
Important Points To Remember
The diagonals of a parallelogram are equal if and only if it is a rectangle.
If a diagonal of a parallelogram bisects one of the angles of the parallelogram then it also bisects the opposite angle.
In a parallelogram, the bisectors of any two consecutive angles intersect at a right angle.
The angle bisectors of a parallelogram form a rectangle.
Mid Point Theorem
A line segment joining the mid points of any two sides of a triangle is parallel to the third side and length of the line segment is half of the parallel side.
Converse Of Mid Point Theorem
A line through the midpoint of a side of a triangle parallel to another side bisects the third side.
Intercept Theorem
If there are three parallel lines and the intercepts made by them on one transversal are equal then the intercepts on any other transversal are also equal.
Angle Sum Property of a Quadrilateral
The sum of the four angles of a quadrilateral is 360°
If we draw a diagonal in the quadrilateral, it divides it into two triangles.
And we know the angle sum property of a triangle i.e. the sum of all the three angles of a triangle is 180°.
The sum of angles of ∆ADC = 180°.
The sum of angles of ∆ABC = 180°.
By adding both we get ∠A + ∠B + ∠C + ∠D = 360°
Hence, the sum of the four angles of a quadrilateral is 360°.
Example
Find ∠A and ∠D, if BC∥ AD and ∠B = 52° and ∠C = 60° in the quadrilateral ABCD.
Solution:
Given BC ∥ AD, so ∠A and ∠B are consecutive interior angles.
So ∠A + ∠B = 180° (Sum of consecutive interior angles is 180°).
∠B = 52°
∠A = 180°- 52° = 128°
∠A + ∠B + ∠C + ∠D = 360° (Sum of the four angles of a quadrilateral is 360°).
∠C = 60°
128° + 52° + 60° + ∠D = 360°
∠D = 120°
∴ ∠A = 128° and ∠D = 120 °.
Types of Quadrilaterals
S No.
Quadrilateral
Property
Image
1.
Kite
a. No Parallel Sides b. Two pairs of adjacent sides are equal.
2.
Trapezium
One pair of opposite sides is parallel.
3.
Parallelogram
Both pairs of opposite sides are parallel.
3.
Rectangle
a. Both the pair of opposite sides are parallel. b. Opposite sides are equal.c. All the four angles are 90°.
4.
Square
a. All four sides are equal. b. Opposite sides are parallel. c. All the four angles are 90°.
5.
Rhombus
a. All four sides are equal. b. Opposite sides are parallel. c. Opposite angles are equal.d. Diagonals intersect each other at the centre and at 90°.
Remark: A square, Rectangle and Rhombus are also a parallelogram.
Properties of a Parallelogram
Theorem 1: When we divide a parallelogram into two parts diagonally then it divides it into two congruent triangles.
∆ABD ≅ ∆CDB
Theorem 2: In a parallelogram, opposite sides will always be equal.
Theorem 3: A quadrilateral will be a parallelogram if each pair of its opposite sides will be equal.
Here, AD = BC and AB = DC
Then ABCD is a parallelogram.
Theorem 4: In a parallelogram, opposite angles are equal.
In ABCD, ∠A = ∠C and ∠B = ∠D
Theorem 5: In a quadrilateral, if each pair of opposite angles is equal, then it is said to be a parallelogram. This is the reverse of Theorem 4.
Theorem 6: The diagonals of a parallelogram bisect each other.
Here, AC and BD are the diagonals of the parallelogram ABCD.
So the bisect each other at the centre.
DE = EB and AE = EC
Theorem 7: When the diagonals of the given quadrilateral bisect each other, then it is a parallelogram.
This is the reverse of the theorem 6.
The Mid-point Theorem
1. If a line segment joins the midpoints of the two sides of the triangle then it will be parallel to the third side of the triangle.
If AB = BC and CD = DE then BD ∥ AE.
2. If a line starts from the midpoint of one line and that line is parallel to the third line then it will intersect the midpoint of the third line.
If D is the midpoint of AB and DE∥ BC then E is the midpoint of AC.
Example
Prove that C is the midpoint of BF if ABFE is a trapezium and AB ∥ EF.D is the midpoint of AE and EF∥ DC.
Solution:
Let BE cut DC at a point G.
Now in ∆AEB, D is the midpoint of AE and DG ∥ AB.
By midpoint theorem, G is the midpoint of EB.
Again in ∆BEF, G is the midpoint of BE and GC∥ EF.
So, by midpoint theorem C is the midpoint of BF.
Hence proved.
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Variations: If the values of two quantities depend on each other in such a way that a change in one causes corresponding change in the other, then the two quantities are said to be in variation. (Scroll down to continue …)
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Direct Variation or Direct Proportion:
Extra:
Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains
constant. That is if
=k [k is a positive number, then x and y are said to vary directly.
In such a case if y1, y2 are the values of y corresponding to the values x1, x of x
respectively then = .
If the number of articles purchased increases, the total cost also increases. More than money deposited in a bank, more is the interest earned.
Quantities increasing or decreasing together need not always be in direct proportion, same in the case of inverse proportion.
When two quantities x and y are in direct proportion (or vary directly), they are
written as
. Symbol
stands for ‘is proportion to’.
Inverse Proportion: Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. That is, if xy
= k, then x and y are said to vary inversely. In this case if y1, y2 are the values of y
corresponding to the values x1, x2 of x respectively then
x1, Y1 = x2, y2 or
=
When two quantities x and y are in inverse proportion (or vary inversely), they are
written as x
. Example: If the number of workers increases, time taken to finish
the job decreases. Or If the speed will increase the time required to cover a given distance decreases.
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Factorisation: Representation of an algebraic expression as the product of two or more expressions is called factorization. Each such expression is called a factor of the given algebraic expression. (Scroll down to continue …)
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When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.
An irreducible factor is a factor which cannot be expressed further as a product of factors.
A systematic way of factorising an expression is the common factor method. It consists of three steps:
Write each term of the expression as a product of irreducible factors
Look for and separate the common factors and
Combine the remaining factors in each term in accordance with the distributive law.
Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping.
In factorisation by regrouping, we should remember that any regrouping (i.e., rearrangement) of the terms in the given expression may not lead to factorisation. We must observe the expression and come out with the desired regrouping by trial and error.
A number of expressions to be factorised are of the form or can be put into the form: a2 + 2ab + b2, a2 – 2ab + b2, a2 – b2 and x2 + (a + b)x + ab. These expressions can be easily factorised using Identities I, II, III and IV
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
a2 – b2 = (a + b) (a – b)
Factorisation
x2 + (a + b)x + ab = (x + a)(x + b)
In expressions which have factors of the type (x + a) (x + b), remember the numerical term gives ab.
Its factors, a and b, should be so chosen that their sum, with signs taken care of, is the coefficient of x.
We know that in the case of numbers, division is the inverse of multiplication. This idea is applicable also to the division of algebraic expressions.
In the case of division of a polynomial by a monomial, we may carry out the division either by dividing each term of the polynomial by the monomial or by the common factor method.
In the case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the divisor polynomial. Instead, we factorise both the polynomials and cancel their common factors.
In the case of divisions of algebraic expressions that we studied in this chapter, we have Dividend = Divisor × Quotient.
In general, however, the relation is Dividend = Divisor × Quotient + Remainder
Thus, we have considered in the present chapter only those divisions in which the remainder is zero.
There are many errors students commonly make when solving algebra exercises.
You should avoid making such errors.
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Graphical presentation of data is easier to understand.
A bar graph is used to show comparison among categories.
A pie graph is used to compare parts of a whole.
A Histogram is a bar graph that shows data in intervals. (Scrol down to continue …)
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Introduction to Graphs
A line graph displays data that changes continuously over periods of time. A line graph which is a whole unbroken line is called a linear graph.
For fixing a point on the graph sheet we need, x-coordinate and y-coordinate.
The relation between dependent variable and
through a graph.
independent variable is shown
A Bar Graph: A pictorial representation of numerical data in the form of bars (rectangles) of uniform width with equal spacing. The length (or height) of each bar
represents the given number.
A Pie Graph: A pie graph is used to compare parts of a whole. The various
observations or components are represented by the sectors of the circle.
A Histogram: Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars (rectangles) show the frequency of the class interval, but there is no gap between the bars as there is no gap between the
class intervals.
Linear Graph: A line graph in which all the line segments form a part of a single line. Coordinates: A point in Cartesian plane is represented by an ordered pair of numbers.
Ordered Pair: A pair of numbers written in specified order.
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Data Handling: Deals with the process of collecting data, presenting it and getting result.
Data mostly available to us in an unorganised form is called raw data. (Scroll down to continue …)
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Grouped data can be presented using histogram. Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars show the frequency of the class interval. Also, there is no gap between the bars as there is no gap between the class intervals.
In order to draw meaningful inferences from any data, we need to organise the data systematically.
Frequency gives the number of times that a particular entry occurs.
Raw data can be ‘grouped’ and presented systematically through ‘grouped frequency distribution’.
Statistics: The science which deals with the collection, presentation, analysis and interpretation of numerical data.
Observation: Each entry (number) in raw data.
Range: The difference between the lowest and the highest observation in a given data.
Array: Arranging raw data in ascending or descending order of magnitude. Data can also presented using circle graph or pie chart. A circle graph shows the relationship between a whole and its part.
There are certain experiments whose outcomes have an equal chance of occurring. A random experiment is one whose outcome cannot be predicted exactly in advance. Outcomes of an experiment are equally likely if each has the same chance of occurring.
Frequency: The number of times a particular observation occurs in the given data.
Class Interval: A group in which the raw data is condensed.
(i) Continuous: The upper limit of a class interval coincides with the lower limit of the next class.
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Square: Number obtained when a number is multiplied by itself. It is the number raised to the power 2. 22 = 2 x 2=4(square of 2 is 4).
If a natural number m can be expressed as n2, where n is also a natural number, then m is a square number.(Scroll down to continue …)
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All square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place. Square numbers can only have even number of zeros at the end. Square root is the inverse operation of square.
There are two integral square roots of a perfect square number.
Positive square root of a number is denoted by the symbol For example, 32=9 gives
Perfect Square or Square number: It is the square of some natural number. If m=n2, then m is a perfect square number where m and n are natural numbers. Example: 1=1 x 1=12, 4=2 x 2=22.
Properties of Square number:
A number ending in 2, 3, 7 or 8 is never a perfect square. Example: 152, 1028, 6593 etc.
A number ending in 0, 1, 4, 5, 6 or 9 may not necessarily be a square number. Example: 20, 31, 24, etc.
Square of even numbers are even. Example: 22 = 4, 42=16 etc.
Square of odd numbers are odd. Example: 52 = 25, 92 = 81, etc.
A number ending in an odd number of zeroes cannot be a perferct square. Example: 10, 1000, 900000, etc.
The difference of squares of two consecutive natural number is equal to their sum. (n + 1)2– n2 = n+1+n. Example: 42 – 32 =4 + 3=7. 122– 112 =12+11 =23, etc.
A triplet (m, n, p) of three natural numbers m, n and p is called Pythagorean
triplet, if m2 + n2 = p2: 32 + 42 = 25 = 52
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Cube number: Number obtained when a number is multiplied by itself three times. 23 = 2 x 2 x 2 = 8, 33 = 3 x 3 x 3=27, etc.
Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They
can be expressed as sum of two cubes in two different ways.
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Numbers obtained when a number is multiplied by itself three times are known as cube numbers. For example 1, 8, 27, … etc.
If in the prime factorisation of any number each factor appears three times, then the
number is a perfect cube.
The symbol
denotes cube root. For example
Perfect Cube: A natural number is said to be a perfect cube if it is the cube of some natural number. Example: 8 is perfect cube, because there is a natural number 2 such that 8 = 23, but 18 is not a perfect cube, because there is no natural number whose cube is 18.
The cube of a negative number is always negative.
Properties of Cube of Number:
Cubes of even number are even.
Cubes of odd numbers are odd.
The sum of the cubes of first n natural numbers is equal to the square of their sum.
Cubes of the numbers ending with the digits 0, 1, 4, 5, 6 and 9 end with digits 0, 1, 4, 5, 6 and 9 respectively.
Cube of the number ending in 2 ends in 8 and cube of the number ending in 8 ends in 2.
Cube of the number ending in 3 ends in 7 and cube of the number ending in 7
ends in 3.
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Ratio: Comparing by division is called ratio. Quantities written in ratio have the sameunit. Ratio has no unit. (Scroll down to continue …)
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Equality of two ratios is called proportion. Product of extremes = Product of means Percentage: Percentage means for every hundred. The result of any division in whichthe divisor is 100 is a percentage. The divisor is denoted by a special symbol %, read as percent. Profit and Loss: (i) Cost Price (CP): The amount for which an article is bought. (ii) Selling Price (SP): The amount for which an article is sold. Additional expenses made after buying an article are included in the cost price and are known as overhead expenses. These may include expenses like amount spent onrepairs, labour charges, transportation, etc. Discount is a reduction given on marked price. Discount = Marked Price – Sale Price. Discount can be calculated when discount percentage is given. Discount
Algebraic Expressions and Identities | Speed Notes
Notes For Quick Recap
Expressions are formed from variables and constants.
Constant: A symbol having a fixed numerical value.
Example: 2,, 2.1, etc. (Scroll down to continue …)
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Variable: A symbol which takes various numerical values. Example: x, y, z, etc.
Algebric Expression: A combination of constants and variables connected by the sign
+, -, and is called algebraic expression.
Terms are added to form expressions.
Terms themselves are formed as product of factors.
Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively.
In general, any expression containing one or more terms with non-zero coefficients (and with variables having non- negative exponents) is called a polynomial.
Like terms are formed from the same variables and the powers of these variables are the same, too.
Coefficients of like terms need not be the same.
While adding (or subtracting) polynomials, first look for like terms and add (or subtract) them; then handle the unlike terms.
There are number of situations in which we need to multiply algebraic expressions: for example, in finding area of a rectangle, the sides of which are given as expressions.
Monomial: An expression containing only one term. Example: -3, 4x, 3xy, etc.
Binomial: An expression containing two terms. Example: 2x-3, 4x+3y, xy-4, etc.,
Polynomial: In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative exponents).
A polynomial may contain any number of terms, one or more than one.
A monomial multiplied by a monomial always gives a monomial.
Multiplication of a Polynomial and a monomial:
While multiplying a polynomial by a monomial, we multiply every term in the polynomial by the mononomial.
Trinomial: An expression containing three terms.
Example:
3x+2y+5z, etc.
In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial).
Note that in such multiplication, we may get terms in the product which are like and have to be combined.
An identity is an equality, which is true for all values of the variables in the equality.
On the other hand, an equation is true only for certain values of its variables.
An equation is not an identity.
The following are the standard identities:
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab +b2
(a + b)(a – b) = a2 – b2
(x + a) (x + b) = x2 + (a + b) x + ab
The above four identities are useful in carrying out squares and products of algebraic expressions.
They also allow easy alternative methods to calculate products of numbers and so on.
Coefficients: In the term of an expression any of the factors with the sign of the term is called the coefficient of the product of the other factors.
Terms: Various parts of an algebraic expression which are separated by + and – signs. Example: The expression 4x + 5 has two terms 4x and 5.
Constant Term: A term of expression having no lateral factor.
Like term: The term having the same literal factors. Example 2xy and -4xy are like terms.
(iii) Unlike term: The terms having different literal factors.
Example:
are unlike terms.
and 3xy
Factors: Each term in an algebraic expression is a product of one or more number (s) and/or literals. These number (s) and/or literal (s) are known as the factor of that term. A constant factor is called numerical factor, while a variable factor is known as
a literal factor. The term 4x is the product of its factors 4 and x.
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A statement of equality of two algebraic expressions involving one or more variables. Example: x + 2 = 3
Linear Equation in One variable: The expressions which form the equation that contain single variable and the highest power of the variable in the equation is one. (Scroll down to continue …)
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Linear Equations in One Variable
An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.
The equations we study in Classes VI, VII and VIII are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.
A linear equation may have for its solution any rational number.
An equation may have linear expressions on both sides. Equations that we studied in Classes VI and VII had just a number on one side of the equation.
Just as numbers, variables can, also, be transposed from one side of the equation to the other.
Occasionally, the expressions forming equations have to be simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.
The utility of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, combination of currency notes, and so on can be solved
using linear equations.
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