Class 10 Exam > Class 10 Videos > Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths
Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths Video Lecture - Class 10
FAQs on Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths Video Lecture - Class 10
| 1. What is the relation between the zeroes and the coefficient of a quadratic polynomial? |  |
Ans. The relation between the zeroes and the coefficient of a quadratic polynomial can be determined using Vieta's formulas. For a quadratic polynomial in the form of ax^2 + bx + c, the sum of the zeroes is equal to -b/a and the product of the zeroes is equal to c/a.
| 2. How can I find the zeroes of a quadratic polynomial if I know the coefficients? | |
Ans. To find the zeroes of a quadratic polynomial with known coefficients, you can use the quadratic formula. The zeroes can be calculated using the formula x = (-b ± √(b^2 - 4ac))/(2a), where a, b, and c are the coefficients of the quadratic polynomial.
| 3. Can a quadratic polynomial have only one zero? | |
Ans. No, a quadratic polynomial cannot have only one zero. A quadratic polynomial always has either two distinct zeroes or no real zeroes. If the discriminant (b^2 - 4ac) is positive, then the polynomial has two distinct real zeroes. If the discriminant is zero, then the polynomial has two equal real zeroes. If the discriminant is negative, then the polynomial has no real zeroes.
| 4. Is it possible for a quadratic polynomial to have complex zeroes? | |
Ans. Yes, it is possible for a quadratic polynomial to have complex zeroes. If the discriminant (b^2 - 4ac) is negative, then the polynomial has no real zeroes. In this case, the zeroes are complex conjugates of each other. Complex conjugate zeroes are of the form a + bi and a - bi, where a and b are real numbers and i is the imaginary unit (√(-1)).
| 5. How are the zeroes of a quadratic polynomial related to its graph? | |
Ans. The zeroes of a quadratic polynomial are related to its graph in the sense that they represent the x-intercepts or the points where the graph intersects the x-axis. If the zeroes are real and distinct, the graph of the polynomial will intersect the x-axis at two different points. If the zeroes are real and equal, the graph will touch the x-axis at a single point. If the zeroes are complex conjugates, the graph will not intersect or touch the x-axis.
Text Transcript from Video
hello friends this video polynomials
part 9 is brought to you by exam viacom
no more fear from exam before watching
this video please make sure that you
have watched formals part 1 - part 8
let's try to derive a relationship
between zeros and coefficients from the
quality problems so this is my quadratic
polynomial px square plus BX plus C
correct this is the quadratic polynomial
because the highest power is too high
score Wake is true and is not the
general form of quadratic polynomial is
ax plus BX plus C and let's assume that
alpha and beta are the factor of this
quadratic polynomial so we can write
this function as K into X minus alpha
into X minus B I can show you if you
want for example it's the zoom three X
square minus ket 1 is plus nothing this
is my quadratic function of this form X
square plus BX plus C now this I know I
can write at this form 3 into X square
minus 7x plus then again if I factorize
this this becomes 3 into X square minus
is 10 5 into 2 is 10 then 5 minus 5 into
no - tube is 10 and minus 5 plus minus 2
is minus 7 so write this as minus 5x
minus 2x plus then we have learn this in
the factorization videos if not please
watch that so this becomes 3 into become
X into X minus 5 then this becomes minus
J correct this is the form we got this
is nothing but he into X minus alpha and
this is nothing but X minus all right so
what I've done I have written this ax
squared plus BX plus C into K into X
minus L Phi 2 X minus V tau where K is a
constant and
find weed are factors of this or zero of
this point of it right I'll find beta
zero of this cone
I'll find with our Zyliss of this
problem correct now the same thing I can
write in this form X square minus alpha
cos beta X plus alpha beta y I can shape
so you have he 2x minus alpha into X
minus 8 and we have this I can write in
this form here this is X square minus
beta X minus Alpha X plus unfamiliar all
right
this is nothing but K into X square
minus other popular will be back x plus
infinite correct this is what we have
got the same thing I can write in this
form KX square minus alpha plus the KX
plus KL 1 that is this so this is more
DFM like this now if you see that this
equation and this equation are same both
are same right because the same equation
I've made like this so I can see that I
can say in this way simple Garcia I can
see that a by K that is a PI K the
coefficient of x square ratio of
coefficient of x squared by D by minus K
into alpha Velveeta the ratio
coefficient of x is equal to C by K by
alpha right what I done since the
equations are see I am saying the ratio
the coefficients of X square X is X
square this is X this is constant the
ratio is same all right so you take
these two what we get is alpha cos beta
is nothing but minus B why the ski and K
cancel and if you take these two this
and
is what we'll get is alpha and beta is
CY right but we have done you take these
two here k cancel reading it alpha plus
beta is equal to minus P by this D will
come here and if you take this and this
here also a handsome what you'll get
will cancel K from volume on alpha beta
is nothing but CY so what we have got
unfurls Whedon is going to minus B Y and
alpha beta is going to C by where alpha
and beta are nothing but zeros of this
quadratic polynomial and a we are see a
efficient of this quadratic point of
aBCL coefficient of this particle thus
we have derived a relationship between
zeros and coefficient of the quadratic
polynomial correct alpha and beta are
zeros and a B and C are coefficient of
quadratic please make a note of this
this is a very useful relationship to
use this alpha plus beta is equal to
minus B by alpha beta is equal to C why
we have alpha and beta zeros of the
quadratic polynomial and a B and C are
nothing but coefficient of the quadratic
one now if I take some examples of 0 the
coefficient of the quadratic element
let's assume this is the function 2x
squared minus 8x plus 6 you can write
this as nothing but u into X square
minus over X plus Q right the same thing
I can factorize this this is minus 4
minus 4 I can write it is nothing but
minus 3 plus minus 1 and minus 3 to
minus 3 so I can write this is nothing
but your constant x squared minus 3x
minus X plus Q all right
they were who into X minus 3 into
x- cut this i have this one
thus i can say that alpha and beta in my
case is nothing but three and one
because this is nothing but K into X
minus alpha into X minus beta correct so
comparing these two equation against say
alpha and V dot V gamma correct also we
know that alpha plus beta is greater
than minus B Y and alpha beta is going
to see why it proved this actually now
first let's find alpha beta since we
know alpha and beta we know that alpha
plus beta is nothing but 3 plus 1 that
is 4 correct alphabetize element in
other words for and now let's try to
find the value minus V Y minus V is
nothing but minus of minus a minus of
minus a by a is 2 that is minus this
will become plus integral school so we
have seen polls where before correct
because this term we got from actual
values of alpha and beta in this form
minus B by we got some formula so we can
say that this formula is true correct
similarly for alpha beta or alpha beta
alpha is nothing but 3 into 1 Ministry
and C by a is nothing but 6 by 2 that is
3 and we can see that he is good to 3 so
we can say that alpha beta is bedroom CY
and this here also we have to that the
formula should be up there I described
it's alpha plus beta is equal to minus B
by 8 alpha beta is equal to C 1 a where
alpha and beta are zeros of this
quadratic polynomial and ABC are
cohesion on the particle this is thus
for example to use the theorem which we
have done design the last slide
alpha plus beta squared minus B my alpha
which is given by you take for example
we have to find the quadratic polynomial
the sum and the product of two zeros are
phone and mirthless technically that is
we are told that alpha + beta + 4 +
alpha beta is ready we are poor and we
have to find a quadratic formula we know
that a quadratic form polynomial formula
is X into K 2 X - 1 5 4 x - VT or
actually we can write this as 1 K into X
square minus alpha plus beta 1 X plus
alpha meter this is the general form
equation where we know alpha plus beta
amino alpha beta all right now we just
put the values so this becomes he into x
squared minus 4x because then for the
Vita is 4 plus alpha beta is 1 since
correct now since this is the value we
have got X square minus alpha plus beta
X plus alpha another X square minus 4x
plus now we can put any value of Q so if
you put K is equal to 1 this equation
becomes X square minus 4x plus P if you
put K is equal to 2 this equation
becomes 2x squared minus 8x plus T if
you put K is going to 3 this equation
becomes X square minus 12 plus 3 all of
these are valid answers
activate so you can be mainly solution
you keep on giving the values of K you
get answer but in general is passed in
exam just put the value of K is equal to
1 and good answer but logically all
these are answers correct because here
also if you see in all these equations
the sum and the product up or and 1
respectively
all right so they can be many solution
for this answers for this question but
in exam
we generally put K squared - 1 look at
that what we have done we have just used
the formula the formula that we have is
for general form of quadratic polynomial
is K into X square minus alpha plus beta
X plus alpha beta and we knew the value
of alpha plus beta and we know the value
of alpha beta we are just substitute the
value of alpha plus beta is 4 and then 4
beta is 1 you create this equation and
then we have put the value of K is equal
to 1 to get this equation 1 K is equal
to 2 we get equation 2 they can keep
putting the values of K different values
you get dependence let's take an example
we have to find again 0 the coefficient
of the quadratic polynomial this is a
polynomial we have 6 square minus 3
minus 7x I can rewrite this as 6 squared
minus 7x minus 3 it is always good to
have spawning in the descending order of
the power you move on 0 so now next
again factorize this first you find that
values of all fine beta and then we'll
prove the same care those find G goes
what will attract eyes this so this is 6
into 3 is 18 and 18 our gate is my need
to do so I'll write like this this is
nothing but is nothing but 6x squared
minus 9x plus 2x minus 2y minus 9 plus Q
is minus M minus 9 into 2 is minus 18
that's what many people you have learned
is in a tradition if not please watch
that P this becomes if you see 3 is
common here P X is common here so this
becomes 2x minus 3 and there becomes 1
this is the value we have got it and
this is nothing but Y now from this you
know that the zeros are those values for
this px good to see right so for the
values that is 3x plus 1 into 2 X minus
3 is equal to 0 for those values of X
you say the other than 0 other why not
me this become 0 that expert 2 minus 1
by 3
or three-way these are the two zeros of
this polynomial function right minus 1
by 3 and T right whether 0 this point in
the function so we know I'll find me
I'll find beta is nothing but minus 1 by
3 and correctly so we have found as ears
now let's again through the formula gish
we had so the formula says alpha plus
beta is equal to minus B my correct so
let's find out forward without order
beta is nothing but minus 1 by 3 plus 3
but that comes out to me
6 common minus 1 plus minus 1 plus 3 3 9
this becomes actually minus 2
why because this is the common way of
the entry into place this so this
becomes 7 by 6 correct but we have to
get 6 as I see in here so we got a
couple of it has been decided by selves
and the value of minus B by is the
finger here - of - spin by a 6 in
several vessels as we have seen both are
equal alpha plus beta is equal to minus
B way
similarly alpha into beta is nothing but
alpha D beta is minus 1 by 3 into 3 by 2
that is minus 1 by 2
it also C by a VM define C Y is nothing
but minus 3 by 6 this - we may be also
we see both the values are same let us
calculate the details but you see why
the here as we see that alpha cos beta
spread to my - anyway and alpha be like
nothing but alpha cos beta is equal to
minus B by and alpha it is going to see
you got a so it passed one of the zeros
zeros are minus 1 by 3 and 3 by 2 and
coefficients a 6b 7 and C's sorry B is
minus m and C's - thank you
visit exam viacom to watch free
education videos try it free on my debts
in the best quality study
careers study from the best tutors and
mentors and much more thanks and serene
part 9 is brought to you by exam viacom
no more fear from exam before watching
this video please make sure that you
have watched formals part 1 - part 8
let's try to derive a relationship
between zeros and coefficients from the
quality problems so this is my quadratic
polynomial px square plus BX plus C
correct this is the quadratic polynomial
because the highest power is too high
score Wake is true and is not the
general form of quadratic polynomial is
ax plus BX plus C and let's assume that
alpha and beta are the factor of this
quadratic polynomial so we can write
this function as K into X minus alpha
into X minus B I can show you if you
want for example it's the zoom three X
square minus ket 1 is plus nothing this
is my quadratic function of this form X
square plus BX plus C now this I know I
can write at this form 3 into X square
minus 7x plus then again if I factorize
this this becomes 3 into X square minus
is 10 5 into 2 is 10 then 5 minus 5 into
no - tube is 10 and minus 5 plus minus 2
is minus 7 so write this as minus 5x
minus 2x plus then we have learn this in
the factorization videos if not please
watch that so this becomes 3 into become
X into X minus 5 then this becomes minus
J correct this is the form we got this
is nothing but he into X minus alpha and
this is nothing but X minus all right so
what I've done I have written this ax
squared plus BX plus C into K into X
minus L Phi 2 X minus V tau where K is a
constant and
find weed are factors of this or zero of
this point of it right I'll find beta
zero of this cone
I'll find with our Zyliss of this
problem correct now the same thing I can
write in this form X square minus alpha
cos beta X plus alpha beta y I can shape
so you have he 2x minus alpha into X
minus 8 and we have this I can write in
this form here this is X square minus
beta X minus Alpha X plus unfamiliar all
right
this is nothing but K into X square
minus other popular will be back x plus
infinite correct this is what we have
got the same thing I can write in this
form KX square minus alpha plus the KX
plus KL 1 that is this so this is more
DFM like this now if you see that this
equation and this equation are same both
are same right because the same equation
I've made like this so I can see that I
can say in this way simple Garcia I can
see that a by K that is a PI K the
coefficient of x square ratio of
coefficient of x squared by D by minus K
into alpha Velveeta the ratio
coefficient of x is equal to C by K by
alpha right what I done since the
equations are see I am saying the ratio
the coefficients of X square X is X
square this is X this is constant the
ratio is same all right so you take
these two what we get is alpha cos beta
is nothing but minus B why the ski and K
cancel and if you take these two this
and
is what we'll get is alpha and beta is
CY right but we have done you take these
two here k cancel reading it alpha plus
beta is equal to minus P by this D will
come here and if you take this and this
here also a handsome what you'll get
will cancel K from volume on alpha beta
is nothing but CY so what we have got
unfurls Whedon is going to minus B Y and
alpha beta is going to C by where alpha
and beta are nothing but zeros of this
quadratic polynomial and a we are see a
efficient of this quadratic point of
aBCL coefficient of this particle thus
we have derived a relationship between
zeros and coefficient of the quadratic
polynomial correct alpha and beta are
zeros and a B and C are coefficient of
quadratic please make a note of this
this is a very useful relationship to
use this alpha plus beta is equal to
minus B by alpha beta is equal to C why
we have alpha and beta zeros of the
quadratic polynomial and a B and C are
nothing but coefficient of the quadratic
one now if I take some examples of 0 the
coefficient of the quadratic element
let's assume this is the function 2x
squared minus 8x plus 6 you can write
this as nothing but u into X square
minus over X plus Q right the same thing
I can factorize this this is minus 4
minus 4 I can write it is nothing but
minus 3 plus minus 1 and minus 3 to
minus 3 so I can write this is nothing
but your constant x squared minus 3x
minus X plus Q all right
they were who into X minus 3 into
x- cut this i have this one
thus i can say that alpha and beta in my
case is nothing but three and one
because this is nothing but K into X
minus alpha into X minus beta correct so
comparing these two equation against say
alpha and V dot V gamma correct also we
know that alpha plus beta is greater
than minus B Y and alpha beta is going
to see why it proved this actually now
first let's find alpha beta since we
know alpha and beta we know that alpha
plus beta is nothing but 3 plus 1 that
is 4 correct alphabetize element in
other words for and now let's try to
find the value minus V Y minus V is
nothing but minus of minus a minus of
minus a by a is 2 that is minus this
will become plus integral school so we
have seen polls where before correct
because this term we got from actual
values of alpha and beta in this form
minus B by we got some formula so we can
say that this formula is true correct
similarly for alpha beta or alpha beta
alpha is nothing but 3 into 1 Ministry
and C by a is nothing but 6 by 2 that is
3 and we can see that he is good to 3 so
we can say that alpha beta is bedroom CY
and this here also we have to that the
formula should be up there I described
it's alpha plus beta is equal to minus B
by 8 alpha beta is equal to C 1 a where
alpha and beta are zeros of this
quadratic polynomial and ABC are
cohesion on the particle this is thus
for example to use the theorem which we
have done design the last slide
alpha plus beta squared minus B my alpha
which is given by you take for example
we have to find the quadratic polynomial
the sum and the product of two zeros are
phone and mirthless technically that is
we are told that alpha + beta + 4 +
alpha beta is ready we are poor and we
have to find a quadratic formula we know
that a quadratic form polynomial formula
is X into K 2 X - 1 5 4 x - VT or
actually we can write this as 1 K into X
square minus alpha plus beta 1 X plus
alpha meter this is the general form
equation where we know alpha plus beta
amino alpha beta all right now we just
put the values so this becomes he into x
squared minus 4x because then for the
Vita is 4 plus alpha beta is 1 since
correct now since this is the value we
have got X square minus alpha plus beta
X plus alpha another X square minus 4x
plus now we can put any value of Q so if
you put K is equal to 1 this equation
becomes X square minus 4x plus P if you
put K is equal to 2 this equation
becomes 2x squared minus 8x plus T if
you put K is going to 3 this equation
becomes X square minus 12 plus 3 all of
these are valid answers
activate so you can be mainly solution
you keep on giving the values of K you
get answer but in general is passed in
exam just put the value of K is equal to
1 and good answer but logically all
these are answers correct because here
also if you see in all these equations
the sum and the product up or and 1
respectively
all right so they can be many solution
for this answers for this question but
in exam
we generally put K squared - 1 look at
that what we have done we have just used
the formula the formula that we have is
for general form of quadratic polynomial
is K into X square minus alpha plus beta
X plus alpha beta and we knew the value
of alpha plus beta and we know the value
of alpha beta we are just substitute the
value of alpha plus beta is 4 and then 4
beta is 1 you create this equation and
then we have put the value of K is equal
to 1 to get this equation 1 K is equal
to 2 we get equation 2 they can keep
putting the values of K different values
you get dependence let's take an example
we have to find again 0 the coefficient
of the quadratic polynomial this is a
polynomial we have 6 square minus 3
minus 7x I can rewrite this as 6 squared
minus 7x minus 3 it is always good to
have spawning in the descending order of
the power you move on 0 so now next
again factorize this first you find that
values of all fine beta and then we'll
prove the same care those find G goes
what will attract eyes this so this is 6
into 3 is 18 and 18 our gate is my need
to do so I'll write like this this is
nothing but is nothing but 6x squared
minus 9x plus 2x minus 2y minus 9 plus Q
is minus M minus 9 into 2 is minus 18
that's what many people you have learned
is in a tradition if not please watch
that P this becomes if you see 3 is
common here P X is common here so this
becomes 2x minus 3 and there becomes 1
this is the value we have got it and
this is nothing but Y now from this you
know that the zeros are those values for
this px good to see right so for the
values that is 3x plus 1 into 2 X minus
3 is equal to 0 for those values of X
you say the other than 0 other why not
me this become 0 that expert 2 minus 1
by 3
or three-way these are the two zeros of
this polynomial function right minus 1
by 3 and T right whether 0 this point in
the function so we know I'll find me
I'll find beta is nothing but minus 1 by
3 and correctly so we have found as ears
now let's again through the formula gish
we had so the formula says alpha plus
beta is equal to minus B my correct so
let's find out forward without order
beta is nothing but minus 1 by 3 plus 3
but that comes out to me
6 common minus 1 plus minus 1 plus 3 3 9
this becomes actually minus 2
why because this is the common way of
the entry into place this so this
becomes 7 by 6 correct but we have to
get 6 as I see in here so we got a
couple of it has been decided by selves
and the value of minus B by is the
finger here - of - spin by a 6 in
several vessels as we have seen both are
equal alpha plus beta is equal to minus
B way
similarly alpha into beta is nothing but
alpha D beta is minus 1 by 3 into 3 by 2
that is minus 1 by 2
it also C by a VM define C Y is nothing
but minus 3 by 6 this - we may be also
we see both the values are same let us
calculate the details but you see why
the here as we see that alpha cos beta
spread to my - anyway and alpha be like
nothing but alpha cos beta is equal to
minus B by and alpha it is going to see
you got a so it passed one of the zeros
zeros are minus 1 by 3 and 3 by 2 and
coefficients a 6b 7 and C's sorry B is
minus m and C's - thank you
visit exam viacom to watch free
education videos try it free on my debts
in the best quality study
careers study from the best tutors and
mentors and much more thanks and serene
About this Video
4.61/5
Rating
Oct 17, 2025
Last updated
Related Exams
Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths for Class 102025 is part of Class 10 preparation. The notes and questions for Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths
have been prepared according to the Class 10 exam syllabus. Information about Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths covers all important topics
for Class 102025 Exam. Find important definitions, questions, notes, meanings, examples, exercises
and tests below for Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths.
Introduction of Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths in English is available as part of our Class 10 preparation & Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths in Hindi
for Class 10 courses. Download more important topics, notes, lectures and mock test series for Class 10 Exam by
signing up for free.
Description
Video Lecture & Questions for Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths Video Lecture - Class 10 - Class 10 full syllabus preparation | Free video for Class 10 exam.
Information about Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths
Here you can find the Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths defined & explained in the simplest way possible. Besides explaining types of Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths
theory, EduRev gives you an ample number of questions to practice Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths tests, examples and also practice Class 10 tests.
Related Searches
Sample Paper
,Maths Video Lecture - Class 10
,Free
,Previous Year Questions with Solutions
,Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials
,mock tests for examination
,ppt
,past year papers
,Viva Questions
,Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials
,study material
,practice quizzes
,Maths Video Lecture - Class 10
,Exam
,Extra Questions
,MCQs
,Summary
,Important questions
,Maths Video Lecture - Class 10
,Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials
,Objective type Questions
,Semester Notes
,shortcuts and tricks
,video lectures
;
Study Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths on the App
Students of Class 10 can study Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Relation between zeroes & coefficient quadratic polynomials (Part -9) - Polynomials, Maths is prepared as per the latest Class 10 syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup