Without actual division prove that 4x4 2x3 6x2 x 1 is exactly divisible by 2x2 x 1

Without actual division prove that x4 + 2x3 - 2x2 + 2x - 3 is exactly divisible by x2 + 2x - 3. 


x2 + 2x - 3
= x2 + 3x - x - 3
= x(x + 3) - 1 (x + 3)
= (x + 3) (x - 1)
Let p(x) = x4 + 2x3 - 2x2 + 2x - 3
We see that
p(-3) = (-3)4 + 2(-3)3 - 2(-3)2 + 2(-3) - 3
= 81 - 54 - 18 - 6 - 3
= 0
Hence by converse of factor theorem, (x + 3) is a factor of p(x).
Also, we see that
p(1) = (1)4 + 2(1)3 - 2(1)2 + 2(1) - 3
= 0
Hence by converse of factor theorem, (x - 1) is a factor of p(x).
From above, we see that
(x + 3) (x - 1), i.e., x2 + 2x - 3 is a factor of p(x)
⇒ p(x) is exactly divisible by (x2 + 2x - 3).


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Brief Contents
Contents
Preface
Chapter Insights
Series Chapter Flow
Class 7
Class 9
Class 8
Class 10
Chapter 1 Number Systems
REMEMBER
KEY IDEAS
INTRODUCTION
EUCLID’S DIVISION LEMMA
FUNDAMENTAL THEOREM OF ARITHMETIC
IRRATIONAL NUMBERS
RATIONAL NUMBERS
PRACTICE QUESTIONS
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Chapter 2 Polynomialsand Rational Expressions
REMEMBER
kEY IDEAS
INTRODUCTION
POLYNOMIAL OF n TH DEGREE
HCF of Given Polynomial
Method for Finding HCF of the Given Polynomials
LCM of the Given Polynomials
Method to Calculate LCM of the Given Polynomials
Relation among the HCF, the LCM and the Product of the Polynomials
Rational Expressions
Rational Expressions in Lowest Terms
Addition/Subtraction of Rational Expressions
Multiplication of Rational Expressions
Division of Rational Expressions
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Chapter 3 LinearEquations inTwo Variables
REMEMBER
KEY IDEAS
INTRODUCTION
ALGEBRAIC EXPRESSIONS
EQUATION
Linear Equation
Simple Equation
Solving an Equation in One Variable
Transposition
Simultaneous Linear Equations
Elimination by Cancellation
Elimination by Substitution
Adding Two Equations and Subtracting One Equation from the Other
Cross-multiplication Method
Graphical Method
Nature of Solutions
Word Problems and Application of Simultaneous Equations
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Chapter 4 Quadratic Equations and Inequalities
REMEMBER
KEY IDEAS
INTRODUCTION
ROOTS OF THE EQUATION
Finding the Roots by Factorization
Consider Equation (1)
Consider Equation (2)
Consider Equation (3)
Consider Equation (4)
Finding the Roots by Using the Formula
Sum and Product of Roots of a Quadratic Equation
Nature of the Roots
Signs of the Roots
CONSTRUCTING A QUADRATIC EQUATION
Constructing a New Quadratic Equation by Changing the Roots of a Given Quadratic Equation
Finding the Roots of a Quadratic Equationby Graphical Method
First Method
Second Method
Equations of Higher Degree
Maximum or Minimum Value of a Quadratic Expression
QUADRATIC INEQUATIONS
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Chapter 5 Statements
REMEMBER
KEY IDEAS
INTRODUCTION
STATEMENT
Truth Value
Negation of a Statement
Compound Statement
Conjunction
Disjunction
Implication or Conditional
Bi-conditional or Bi-implication
Tautology
Contingency
Logically Equivalent Statements
Laws of Algebra of Statements
List of Equivalences Based on Implications
OPEN SENTENCE
Quantifiers
Universal Quantifier
Existential Quantifier
Negation of Statements Involving Quantifiers
METHODS OF PROOF
Direct Proof
Indirect Proof
METHODS OF DISPROOF
Counter Example Method
Method of Contradiction
APPLICATION TO SWITCHING NETWORKS
Switching Network
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Chapter 6 Sets, Relations and Functions
REMEMBER
KEY IDEAS
INTRODUCTION
SET
Elements of a Set
Some Sets of Numbers and Their Notations
Cardinal Number of a Set
Representation of Sets
Roster or List Method
Set-builder Method
Types of Sets
Singleton Set
Equivalent Sets
Equal Sets
Disjoint Sets
Subset and Superset
Proper Subset
Power Set
Universal Set
Complement of a Set
Operations on Sets
Union of Sets
Intersection of Sets
Difference of Sets
Symmetric Difference of Sets
Dual of an Identity
Venn Diagrams
Some Formulae on the Cardinality of Sets
ORDERED PAIR
Cartesian Product of Sets
Some Results on Cartesian Product
Representation of A × B using Arrow Diagram
Representation of A × B Using a Tree Diagram
Graphical Representation of A × B
RELATION
Definition
Domain and Range of a Relation
Representation of Relations
Roster Method (or) List Method
Set-builder Method
Arrow Diagram
Inverse of a Relation
Types of Relations
Properties of Relations
Identity Relation
FUNCTION
Domain and Co-domain
Range
Arrow Diagram
Difference Between Relations and Functions
Types of Functions
One-one Function or Injection
Many to One Function
Onto Function or Surjection
Into Function
Bijective Function
Inverse of a Function
Identity Function
Constant Function
Equal Functions
Composite Function (or) Product Function
Real Function
Even and Odd Functions
Graphs of Functions
Zeroes of a Function
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Chapter7 Progressions
REMEMBER
KEY IDEAS
INTRODUCTION
SEQUENCE
Finite and Infinite Sequences
Series
Arithmetic Progression (AP)
Arithmetic Mean (AM)
Some Important Results
GEOMERIC PROGRESSION (GP)
Infinite Geometric Progression
HARMONIC PROGRESSION (HP)
Harmonic Mean (HM)
Inserting n Harmonic Means Between Two Numbers
Relation between AM, HM and GM of Two Numbers
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Chapter 8 Trigonometry
REMEMBER
KEY IDEAS
INTRODUCTION
ANGLE
Systems of Measurement of Angle
Sexagesimal System
Centesimal System
Circular System
Relation Between the Units of the Three Systems
TRIGONOMETRIC RATIOS
Trigonometric Ratios of Compound Angles
Standard Position of the Angle
Co-terminal Angles
Signs of Trigonometric Ratios
Trigonometric Ratios of (90° − θ)
Trigonometric Ratios of (90° + θ)
Trigonometric Ratios of (180° − θ)
Trigonometric Ratios of (180° + θ)
Trigonometric Tables
HEIGHTS AND DISTANCES
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Chapter 9 Limits
REMEMBER
KEY IDEAS
INTRODUCTION
Limit of a Function
Meaning of ‘x → a’
Method of Substitution
Method of Factorization
Method of Rationalization
Using the Formula
Limits as x Tends to Infinity
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Chapter 10 Matrices
REMEMBER
KEY IDEAS
INTRODUCTION
ORDER OF A MATRIX
Various Types of Matrices
Rectangular Matrix
Row Matrix
Column Matrix
Null Matrix or Zero Matrix
Square Matrix
Diagonal Matrix
Scalar Matrix
Identity Matrix or Unit Matrix
Comparable Matrices
Equality of Two Matrices
OPERATIONS ON MATRICES
Multiplication of a Matrix by a Scalar
Addition of Matrices
Properties of Matrix Addition
Matrix Subtraction
Transpose of a Matrix
Symmetric Matrix
Skew-symmetric Matrix
Multiplication of Matrices
Properties of Matrix Multiplication
DETERMINANT
Singular Matrix
Non-singular Matrix
Multiplicative Inverse of a Square Matrix
Multiplicative Inverse of a 2 × 2 Square Matrix
Method for Finding Inverse of a 2 × 2 Square Matrix
Solution of Simultaneous Linear Equations in Two Variables
Matrix Inversion Method
Cramer’s Rule or Cramer’s Method
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Chapter 11 Remainderand Factor Theorems
REMEMBER
KEY IDEAS
INTRODUCTION
REMAINDER THEOREM
FACTOR THEOREM
Factorization of Polynomials Using Factor Theorem
Horner’s Process for Synthetic Division of Polynomials
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Chapter 12 Statistics
REMEMBER
KEY IDEAS
INTRODUCTION
Data
Types of Data
Raw Data
Grouped Data
Some Basic Definitions
Tabulation or Presentation of Data
Class Interval
Class Boundaries
Class Size
Class Mark or Mid-value
Statistical Graphs
Bar Graph
Histograms
Remarks
Important Observations
Frequency Polygon
Frequency Curve
Cumulative Frequency Curves
Measures of Central Tendencies for Ungrouped Data
Arithmetic Mean or Mean (AM)
Weighted Arithmetic Mean
Some Important Results About AM
Median
Some Important Facts About Median
Mode
Properties of Mode
Empirical Relationship Among Mean, Median, and Mode
Measure of Central Tendencies for Grouped Data
Mean of Grouped Data
Short-cut Method for Finding the Mean of Group Data (Deviation Method)
Median of Grouped Data
Mode of Grouped Data
Range
Calculation of Variance and Standard Deviation for Raw Data
Calculation of Variance and SD for a Grouped Data
Coefficient of Variation (CV)
Quartiles
First Quartile or Lower Quartile
Third Quartile or Upper Quartile
Semi-inter Quartile Range or Quartile Deviation (QD)
Estimation of Median and Quartiles from Ogive
Estimation of Mode from Histogram
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Chapter 13 Geometry
REMEMBER
KEY IDEAS
INTRODUCTION
SYMMETRY
Line Symmetry
Point Symmetry
Illustrations
Image of a Point About a Line
Image of a Line Segment about a Line
Image of an Angle About a Line
SIMILARITY
Similarity of Geometrical Figures
Similarity as a Size Transformation
Enlargement
Properties of Size Transformation
Model
Map
Criteria for Similarity of Triangles
Similar Triangles
Three Similarity Axioms for Triangles
Right Angle Theorem
Pythagoras Theorem
Appolonius Theorem
Converse of Basic Proportionality Theorem
Vertical Angle Bisector Theorem
Converse of Vertical Angle Bisector Theorem
Concurrency—Geometric Centres of a Triangle
Circum-centre
In-centre
Ortho-centre
Centroid
Some Important Points
CIRCLES
Chord
Theorem 1
Properties of Chords and Related Theorems
Theorem 2
Theorem 3
Angles Subtended by Equal Chords at the Centre
Theorem 4
Angles Subtended by an Arc
Property 1
Property 2
Cyclic Quadrilateral
Theorem 5
Theorem 6
Tangents
Theorem 7
Converse of the Theorem
Theorem 8
Chords
Theorem 9
Alternate Segment and Its Angles
Theorem 10 Alternate Segment Theorem
Converse of Alternate Segment Theorem
Common Tangents to Circles
Common Tangent
Properties of Common Tangents
Constructions Related to Circles
LOCUS
Equation of a Locus
Steps to Find the Equation of a Locus
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Chapter 14 Mensuration
REMEMBER
KEY IDEAS
INTRODUCTION
Circle and Semi-Circle
Circular Ring
Sectors and Segements
Rotations Made by a Wheel
Equilateral Triangle
PRISMS
Lateral Surface Area (LSA) of a Prism
Total Surface Area (TSA) of a Prism
Volume of a Prism
CUBES AND CUBOIDS
Cuboid
Cube
RIGHT CIRCULAR CYLINDER
Hollow Cylinder
PYRAMID
Right Pyramid
CONE
Right Circular Cone
Hollow Cone
Cone Frustum (or a Conical Bucket)
SPHERE
Solid Sphere
Hollow Sphere
Hemisphere
Hemispherical Shell
Formulae to Memorize
Sphere
Spherical Shell/Hollow Sphere
Hemisphere
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Chapter 15 Coordinate Geometry
REMEMBER
KEY IDEAS
INTRODUCTION
COORDINATES OF A POINT
Convention of Signs
POINTS ON THE PLANE
Point on X-axis and Y-axis
Distance Between Two Points
Applications of Distance Formula
STRAIGHT LINES
Inclination of a Line
Slope or Gradient of a Line
Some Results on the Slope of a Line
Theorem 1
Theorem 2
The Slope of a Line Passing Through the Points (x1, y1) and (x2, y2)
Intercepts of a Straight Line
Equation of a Line in General Form
Equations of Some Standard Lines
Oblique Line
Different Forms of Equations of Oblique Lines
Area of Triangle
Area of a Quadrilateral
Section Formulae
Section Formula
Mid-point
Points of Trisection
Centroid
Equation of a Line Parallel or Perpendicular to the Given Line
Median of the Triangle
Altitude of the Triangle
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Capter 16 Mathematical Induction and Binomial Theorem
REMEMBER
KEY IDEAS
INTRODUCTION
THE PRINCIPLE OF MATHEMATICAL INDUCTION
Binomial Expression
Pascal Triangle
Factorial Notation and nCr Representation
Binomial Theorem
Important Inferences from the above Expansion
Binomial Theorem
Middle Terms in the Expansion of (x + y)n
Term Independent of x
The Greatest Coefficient in the Expansion of (1 + x)n (where n is aPositive Integer)
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Chapter 17 Modular Arithmetic
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INTRODUCTION
CONGRUENCE
Set of Residues
Modular Addition
Modular Multiplication
Construction of Caley’s Table
Linear Congruence
Solution of Linear Congruence
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Chapter 18 Linear Programming
REMEMBER
KEY IDEAS
INTRODUCTION
Convex Set
Objective Function
Closed-Convex Polygon
Open-Convex Polygon
The Fundamental Theorem
Feasible Region
Feasible Solution
Optimum Solution
Graphical Method of Solving a Linear Programming Problem
General Graphical Method for Solving Linear ProgrammingProblems
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Chapter 19 Computing
REMEMBER
KEY IDEAS
INTRODUCTION
Historical Development of Computing
Characteristics of a Computer
Architecture of a Computer
Software
Algorithm
Flowchart
Operators
Basic
Constants
Numeric
Alphanumeric
Variables
Numeric variable
Alphanumeric variables or string variables
Rules for Declaration of Variables
Basic Operators
Arithmetic Operators
Logical Operators
Relational Operators
Order in which Arithmetic Operators are Evaluated
Basic Statements
REM
LET
READ … DATA
PRINT
END
STOP
CONDITIONAL STATEMENTS
Branching Statements
IF-THEN Statement
IF-THEN–ELSE
Looping Statements
Unconditional Statements
GO TO
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Chapter 20 Permutations and Combinations
REMEMBER
KEY IDEAS
INTRODUCTION
Sum Rule of Disjoint Counting
General Form of Sum Rule
Product Rule or Multiplication Rule
Generalization of Product Rule
PERMUTATIONS
Factorial Notation
General Formula for Permutations (Repetitions not Allowed)
Explanation
General Formula for Permutations with Repetitions Allowed
COMBINATIONS
General Formula for Combinations
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Chapter 21 Probability
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INTRODUCTION
Sample Space
Event
Probability of an Event
Probability of Non-occurrence of an Event E
Classification of a Pack (or Deck) of Cards
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Chapter 22 Banking
REMEMBER
KEY IDEAS
INTRODUCTION
DEPOSIT ACCOUNTS
Savings Bank Account
Depositing Money in the Bank Accounts
Demand Draft
Withdrawing Money From Saving Bank Account
Types of Cheques
Bearer Cheque
Crossed Cheque
Bouncing of Cheques
Safeguards to be taken while Maintaining ‘Cheque-Operated Accounts’
Parties dealing with a cheque
Drawer
Drawee
Payee
Calculation of Interest on Savings Accounts in Banks
Current Account
Term Deposit Accounts
Fixed Deposit Accounts
Recurring Deposit Accounts
Loans
Demand Loans
Term Loans
Overdrafts (ODs)
Calculation of Interest on Loans
Compound Interest
HIRE PURCHASE AND INSTALMENT SCHEME
Hire Purchase Scheme
Important Features of Hire Purchase Scheme
Instalment Scheme
Finding the Rate of Interest on Buying in Instalment Scheme
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Chapter 23 Taxation
REMEMBER
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INTRODUCTION
INCOME TAX
SALES TAX
Calculation of Sales Tax
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Chapter 24 Instalments
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INTRODUCTION
Cash Price
Initial Payment or Down Payment
Repayment of Loan
Hire Purchase Scheme
Important Features of Hire Purchase Scheme
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Chapter 25 Shares and Dividends
REMEMBER
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INTRODUCTION
CLASSIFICATION OF SHARE CAPITAL
Nominal Value of a Share (NV)
Market Value of a Share (MV)
Dividend
Return on Investment
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Chapter 26 Partial Fractions
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Method 1
Method 2
Method 3
Method 4
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Chapter 27 Logarithms
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KEY IDEAS
INTRODUCTION
Definition
SYSTEM OF LOGARITHMS
Properties
Laws
Variation of logax with x
Sign of logax for Different Values of x and a
To Find the log of a Number to the Base 10
The Rule to Obtain the Characteristic of log x
To Find the log of a Number from the log Tables
ANTILOG
To Find the Antilog
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Is x 4 2x 3 2x 2 2x 3 is exactly divisible by x 2 2x 3?

Hence, X 4 + 2 X 3 + 2 X 2 + 2 X - 3 is exactly divisible by X 2 + 2 X - 3 .

What should be added to F x )= 4x4 2x3 2x2 x 1?

Hence we should add –rx = 61x – 65 to fx so that the resulting polynomial is divisible by g x.

What must be subtracted from 4x 4 2x 3 6x 2 2x 6 so that the result is exactly divisible by 2x 2 x 1?

Hence, −6 should be subtracted from 4x4−2x3−6x2+x−5 so that result is exactly divisible by 2x2+x−1.

What must be added to x4 2x3 2x2 x 1?

Hence, the expression that must be added to x 4 + 2 x 3 - 2 x 2 + x - 1 is to make it exactly divisible by x 2 + 2 x - 3 .