Some definitions
Real polynomial
Complex polynomial
Degree of a polynomial
Polynomial equation
Some Results on roots of an equation
1. An equation of degree n has n roots, real or imaginary.
2. Surd and imaginary roots always occur in pairs, i.e. if 5-3i is a root of an equation, then 5 +3i is also its root. Similarly, if 3+SQRT(5) is a root of a given equation, then 3-SQRT(5) is also its root.
3. An odd degree equation has at least one real root, whose sign is opposite to that of its last term, provided that the coefficient of highest degree term is positive.
4. Every equation of an even degree whose constant term is negative and the coefficient of highest degree term is positive, has at least two real roots, one positive and one negative
3. Position of roots of a polynomial equation
4 Descartes rule of signs
5. Relations between roots and coefficients
6. Formation of a polynomial equation form given roots
7. Transformation of equations
8. Roots of a quadratic equation with real coefficients
ax²+bx+c where a≠0, a,b,c Є R is a quadratic equation with real coefficients.
The quantity D = b²-4ac is the called the discriminant of the quadratic equation.
1. The roots are real and distinct if and only if D>0.
2. The roots are real and equal if and only D = 0
3. The roots are complex with non-zero imaginary part if and only if D<0 .="" br="">4. The roots are rational iff a,b,c are rational and D is a proper square.
5. The roots are of the form p+√q (p,q Є Q), iff a,b,c are rational and D is not a perfrect square.
6. If a =1, b,c ЄI and the roots are rational numbers, then these roots must be integers.
7. If a quadratic equation in x has more than two roots, then it is an identity in x that is a=b=c=o.
9. Quadratic expression and its graph
Graph of a quadratic expression is a parabola.
10. Sign of a quadratic expression for real values of the variable
11. Solution of inequations
12. Position of roots of a quadratic equation
13. Common roots
14. Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions
15. Condition for resolution into linear factors of a quadratic function
16. Algebraic interpretation of Rolle’s theorem
Updated 9 Jan 2016, 7 June 2008
Showing posts with label Quadratic equations. Show all posts
Showing posts with label Quadratic equations. Show all posts
Saturday, January 9, 2016
Wednesday, December 17, 2008
Quadratic Equations and Expressions - Definitions
Real Polynomials: Coefficients are real numbers and variables take real values.
Complex Polynomials: Coefficients are complex numbers and variable is varying complex number.
Polynomial equation:
f(x) = 0
Roots of an equation: The values of the variable satisfying the given equation are called its roots.
Complex Polynomials: Coefficients are complex numbers and variable is varying complex number.
Polynomial equation:
f(x) = 0
Roots of an equation: The values of the variable satisfying the given equation are called its roots.
Some conclusions on Roots of a Polynomial Equation
1. An equation of degree n has n roots, real or imaginary.
2. Surd and imaginary roots always occur in pairs, i.e. if 5-3i is a root of an equation, then 5 +3i is also its root. Similarly, if 3+SQRT(5) is a root of a given equation, then 3-SQRT(5) is also its root.
3. An odd degree equation has at least one real root, whose sign is opposite to that of its last term, provided that the coefficient of highest degree term is positive.
4. Every equation of an even degree whose constant term is negative and the coefficient of highest degree term is positive, has at least two real roots, one positive and one negative
2. Surd and imaginary roots always occur in pairs, i.e. if 5-3i is a root of an equation, then 5 +3i is also its root. Similarly, if 3+SQRT(5) is a root of a given equation, then 3-SQRT(5) is also its root.
3. An odd degree equation has at least one real root, whose sign is opposite to that of its last term, provided that the coefficient of highest degree term is positive.
4. Every equation of an even degree whose constant term is negative and the coefficient of highest degree term is positive, has at least two real roots, one positive and one negative
Descartes Rule of Signs
The maximum number of positive real roots of a polynomial equation f(x) = 0 is the number of changes of signs from positive to negative and negative to positive in f(x).
Relations between roots and coefficients
S1 = α1+α2+...+αn = -a1/a0
S2 = α1α2 + α1α3+... = Σαiαj; i≠j = (-1)² (a2/a0)
Sn = α1α2...αn = (-1)n(const. term/a0) {constant term = an)
S2 = α1α2 + α1α3+... = Σαiαj; i≠j = (-1)² (a2/a0)
Sn = α1α2...αn = (-1)n(const. term/a0) {constant term = an)
Formation of a polynomial equation from given roots
If α1, α2, α3,...,αk are the roots of an nth degree equation
xn-S1xn-1+S2xn-2-S3xn-3++...+(-1)nSn = 0
where Sk denotes the sum of the products of roots taken k at a time.
xn-S1xn-1+S2xn-2-S3xn-3++...+(-1)nSn = 0
where Sk denotes the sum of the products of roots taken k at a time.
Roots of a quadratic equation with real coefficients
ax²+bx+c where a≠0, a,b,c Є R is a quadratic equation with real coefficients.
The quantity D = b²-4ac is the called the discriminant of the quadratic equation.
1. The roots are real and distinct if and only if D>0.
2. The roots are real and equal if and only D = 0
3. The roots are complex with non-zero imaginary part if and only if D<0.
4. The roots are rational iff a,b,c are rational and D is a proper square.
5. The roots are of the form p+√q (p,q Є Q), iff a,b,c are rational and D is not a perfrect square.
6. If a =1, b,c ЄI and the roots are rational numbers, then these roots must be integers.
7. If a quadratic equation in x has more than two roots, then it is an identity in x that is a=b=c=o.
The quantity D = b²-4ac is the called the discriminant of the quadratic equation.
1. The roots are real and distinct if and only if D>0.
2. The roots are real and equal if and only D = 0
3. The roots are complex with non-zero imaginary part if and only if D<0.
4. The roots are rational iff a,b,c are rational and D is a proper square.
5. The roots are of the form p+√q (p,q Є Q), iff a,b,c are rational and D is not a perfrect square.
6. If a =1, b,c ЄI and the roots are rational numbers, then these roots must be integers.
7. If a quadratic equation in x has more than two roots, then it is an identity in x that is a=b=c=o.
Sign of a quadratic expression for real values of the variable
For real values of x, the sign of the quadratic expression f(x) = ax² +bx+c is the same as that of 'a' except when the roots of the equation ax²+bx+c are real and distinct and x lies between them.
ax²+bx+c is greater than 0 for all x Є R iff a is greater than 0 and D is less than zero. (D is discriminant b²-ac), and
ax²+bx+c is lesser than 0 for all x Є R iff a is less than 0 and D is less than zero.
ax²+bx+c is greater than 0 for all x Є R iff a is greater than 0 and D is less than zero. (D is discriminant b²-ac), and
ax²+bx+c is lesser than 0 for all x Є R iff a is less than 0 and D is less than zero.
Friday, November 7, 2008
IT JEE Mathematics Study Guide 7. Quadratic Equations and Expressions - Revision Facilitator
R.D. Sharma Objective Mathematics
7.1 Some definitions and results
7.2 Some results on roots of an equation
7.3 Position of roots of a polynomial equation
7.4 Descartes rule of signs
7.5 Relations between roots and coefficients
7.6 Formation of a polynomial equation from given roots.
7.7 Transformation of equations
7.8 Roots of a quadratic equation with real coefficients
7.9 Quadratic expression and its graph
7.10 Sign of a quadratic expression for real values of the variable
7.11 Solution of inequations
7.12 Position of roots of a quadratic equation
7.13 Common roots
7.14 Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions
7.15 Condition for resolution into linear factors of a quadratic function
7.16 Algebraic interpretation of Rolle’s theorem
Study Plan
Always circle difficult concepts or difficult problems. You need to revise them later on more intensively and make them easy (Any difficult issue becomes easy as you understand the concept and related concepts better.)
Day 1
7.1 Some definitions and results
7.2 Some results on roots of an equation
7.3 Position of roots of a polynomial equation
Ex. 1
7.4 Descartes rule of signs
Ex. 1 to 3
Day 2
7.5 Relations between roots and coefficients
Ex. 1 to 3
7.6 Formation of a polynomial equation from given roots.
7.7 Transformation of equations
Ex. 1 to 4
Day 3
7.8 Roots of a quadratic equation with real coefficients
7.9 Quadratic expression and its graph
7.10 Sign of a quadratic expression for real values of the variable
Day 4
7.11 Solution of inequations
Ex. 1,2 and Ex 1,2
7.12 Position of roots of a quadratic equation
EX. 1 to 4
Day 5
7.13 Common roots
Ex. 1 to 3
Day 6
7.14 Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions
Ex. 1 to 3
7.15 Condition for resolution into linear factors of a quadratic function
Ex. 1 ro 2
Day 7
7.16 Algebraic interpretation of Rolle’s theorem
Ex. 1,2
Illustrative Objective Type Questions: 1 to 10
Day 8
I.O.T.Q.: 11 to 30
Day 9
I.O.T.Q.: 31 to 50
Day 10.
I.O.T.Q.: 51 to 64
Day 11
Objective Type Questions: 1 to 20
Day 12
O.T.P.: 21 to 60 odd numbered questions
(Persons who can do more problems can do more problems)
Day 13
O.T.P.: 61 to 100 even numbered questions
(Persons who can do more problems can do more problems)
Day 14
O.T.P.: 101 to 140 odd numbered questions
(Persons who can do more problems can do more problems)
Day 15
O.T.P.:141 to 184 even numbered questions
(Persons who can do more problems can do more problems)
Revision Period
Day 16
Fill in the blanks exercises: 1 to 20
Day 17
True or false exercises: 1 to 13
Day 18
Practice Exercises: 1 to 10
Day 19
Practice Exercises:11 to 20
Day 20
Practice Exercises: 21 to 30
Day 21
Practice Exercises: 31 to 40
Day 22
Practice Exercises: 41 to 50
Day 23
Practice Exercises: 51 to 60
Day 24
Practice Exercises: 61 to 63
Day 25
Revision of concepts of the chapter
Day 26
O.T.P.: 21 to 60 even numbered questions
Day 27
O.T.P.: 61 to 100 odd numbered questions
Day 28
O.T.P.: 101 to 140 even numbered questions
Day 29
O.T.P.:141 to 184 odd numbered questions
Day 30
Revision of some difficult problems
7.1 Some definitions and results
7.2 Some results on roots of an equation
7.3 Position of roots of a polynomial equation
7.4 Descartes rule of signs
7.5 Relations between roots and coefficients
7.6 Formation of a polynomial equation from given roots.
7.7 Transformation of equations
7.8 Roots of a quadratic equation with real coefficients
7.9 Quadratic expression and its graph
7.10 Sign of a quadratic expression for real values of the variable
7.11 Solution of inequations
7.12 Position of roots of a quadratic equation
7.13 Common roots
7.14 Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions
7.15 Condition for resolution into linear factors of a quadratic function
7.16 Algebraic interpretation of Rolle’s theorem
Study Plan
Always circle difficult concepts or difficult problems. You need to revise them later on more intensively and make them easy (Any difficult issue becomes easy as you understand the concept and related concepts better.)
Day 1
7.1 Some definitions and results
7.2 Some results on roots of an equation
7.3 Position of roots of a polynomial equation
Ex. 1
7.4 Descartes rule of signs
Ex. 1 to 3
Day 2
7.5 Relations between roots and coefficients
Ex. 1 to 3
7.6 Formation of a polynomial equation from given roots.
7.7 Transformation of equations
Ex. 1 to 4
Day 3
7.8 Roots of a quadratic equation with real coefficients
7.9 Quadratic expression and its graph
7.10 Sign of a quadratic expression for real values of the variable
Day 4
7.11 Solution of inequations
Ex. 1,2 and Ex 1,2
7.12 Position of roots of a quadratic equation
EX. 1 to 4
Day 5
7.13 Common roots
Ex. 1 to 3
Day 6
7.14 Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions
Ex. 1 to 3
7.15 Condition for resolution into linear factors of a quadratic function
Ex. 1 ro 2
Day 7
7.16 Algebraic interpretation of Rolle’s theorem
Ex. 1,2
Illustrative Objective Type Questions: 1 to 10
Day 8
I.O.T.Q.: 11 to 30
Day 9
I.O.T.Q.: 31 to 50
Day 10.
I.O.T.Q.: 51 to 64
Day 11
Objective Type Questions: 1 to 20
Day 12
O.T.P.: 21 to 60 odd numbered questions
(Persons who can do more problems can do more problems)
Day 13
O.T.P.: 61 to 100 even numbered questions
(Persons who can do more problems can do more problems)
Day 14
O.T.P.: 101 to 140 odd numbered questions
(Persons who can do more problems can do more problems)
Day 15
O.T.P.:141 to 184 even numbered questions
(Persons who can do more problems can do more problems)
Revision Period
Day 16
Fill in the blanks exercises: 1 to 20
Day 17
True or false exercises: 1 to 13
Day 18
Practice Exercises: 1 to 10
Day 19
Practice Exercises:11 to 20
Day 20
Practice Exercises: 21 to 30
Day 21
Practice Exercises: 31 to 40
Day 22
Practice Exercises: 41 to 50
Day 23
Practice Exercises: 51 to 60
Day 24
Practice Exercises: 61 to 63
Day 25
Revision of concepts of the chapter
Day 26
O.T.P.: 21 to 60 even numbered questions
Day 27
O.T.P.: 61 to 100 odd numbered questions
Day 28
O.T.P.: 101 to 140 even numbered questions
Day 29
O.T.P.:141 to 184 odd numbered questions
Day 30
Revision of some difficult problems
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