Class 8 Exam > Class 8 Videos > Standard Algebraic Identities with Proof
Standard Algebraic Identities with Proof Video Lecture - Class 8
FAQs on Standard Algebraic Identities with Proof Video Lecture - Class 8
| 1. What are the standard algebraic identities? |  |
Ans. The standard algebraic identities include:
1. $(a + b)^2 = a^2 + 2ab + b^2$
2. $(a - b)^2 = a^2 - 2ab + b^2$
3. $a^2 - b^2 = (a + b)(a - b)$
4. $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$
5. $(a^3 + b^3) = (a + b)(a^2 - ab + b^2)$
| 2. How are the standard algebraic identities derived? | |
Ans. The standard algebraic identities are derived using mathematical manipulations and expansions. For example, to derive the $(a + b)^2$ identity, we start with the expression $(a + b)(a + b)$ and use the distributive property to expand it as $a(a + b) + b(a + b)$. Simplifying further gives us $a^2 + ab + ab + b^2$, which can be written as $a^2 + 2ab + b^2$. Similarly, the other identities can be derived through similar expansions and simplifications.
| 3. Why are the standard algebraic identities important in mathematics? | |
Ans. The standard algebraic identities are important in mathematics because they provide a shortcut for solving complex algebraic expressions. By using these identities, we can simplify expressions, factorize polynomials, and solve equations more easily. They also help in understanding the relationships between different algebraic terms and operations, making it easier to manipulate and solve equations.
| 4. How can the standard algebraic identities be used to solve equations? | |
Ans. The standard algebraic identities can be used to solve equations by simplifying the expressions on both sides of the equation using the identities. By applying the identities, we can manipulate the equation to isolate the variable we want to solve for. This makes solving equations faster and more efficient. Additionally, the identities can also be used to factorize polynomials, which can further simplify equations and aid in finding their solutions.
| 5. Can the standard algebraic identities be used in other branches of mathematics? | |
Ans. Yes, the standard algebraic identities are not limited to algebra alone and can be used in various branches of mathematics. They are extensively used in calculus, trigonometry, and linear algebra, among others. The identities help in simplifying expressions, solving equations, and proving mathematical theorems. They provide a foundation for further mathematical concepts and techniques, making them a fundamental tool in many areas of mathematics.
Text Transcript from Video
now we'll learn how identity is like a
plus B whole square equal to a square
plus 2 a B plus B Square and identity
like a minus B whole square equal to a
square minus 2 a B plus B Square and one
more which is a square minus B Square
equal to a minus B into a plus B can be
derived so now let's talk about the
first identity which is a plus B whole
square we will start from the left hand
side and try to reach towards the
right-hand side of the identity so what
is the left-hand side left-hand side is
a plus B whole square so let's start
from here what is the meaning of a plus
B whole square a plus B whole square is
actually equal to a plus B multiplied to
itself okay
so how we multiply while multiplying two
expressions here the two expressions are
a plus B and a plus B so while
multiplying what we do we take the first
term of the first expression and then we
multiply it to the first term of the
second expression and then again we take
first term of the first expression and
then we multiply it to the second term
of the second expression and if we have
more than two terms we have to continue
this procedure okay then we start with
the second term of the first expression
and repeat the procedure we multiply the
second term of the first expression to
the first term of the second expression
and then we multiplied the second term
of the first section expression to the
second term of the second expression and
if we have more terms we have to repeat
this procedure okay so let's write what
we will get by doing this procedure it
will be equal to first a will get
multiplied to a which will be a
multiplied by a then we have a plus sign
why because this B is positive a is
getting multiplied to B so we will get a
B and now since there were only two
terms in the second expression we will
start with the second term of the first
expression that is B we will multiply
this B by the first term of the second
expression which is a B multiplied by a
is B a and we write it in alphabetical
order that will be a B and then B will
get multiplied to B that is the second
term of the first expression and the
second term of the second expression
both are B B
multiplied by B is like this now let us
solve this further a multiplied by a is
a square a B and a B there are two ABS
so plus 2 a B and then we have plus P
multiplied by B that is B Square so it
is plus B Square so what is the value of
a plus B whole square a plus B whole
square is actually equal to a square
plus 2 a B plus B Square so this was the
derivation of our first identity which
says a plus B whole square is equal to a
square plus 2 a B plus B Square ok now
let's talk about the second identity
second identity is this which says a
minus B whole square is equal to a
square minus 2 a B plus B square so what
I'll do I'll write the left-hand side of
the expression here the left-hand side
of the expression is a minus B whole
square okay so what is the meaning of a
minus B whole square a minus B whole
square is nothing but a minus B
multiplied by itself that is a minus P a
minus B whole square is nothing but a
minus B
multiplied by a minus B again we have to
repeat the procedure of multiplication
this a will get multiplied by this a
first and then it will get multiplied to
this B so this will give us a multiplied
by a is a squared this time I am writing
directly okay and then this a multiplied
by minus B so it will be minus a B why
this minus sign in multiplication if -
get multiplied by minus we will get a
positive term if plus gets multiplied by
plus then also we get a positive term
but if a minus gets multiplied by plus
or a plus gets multiplied by minus we
will get a minus term this we have
learnt in the video where I am
explaining you how to multiply two
algebraic expressions you can refer that
video okay so since this is there so
next term what we have to do we have to
since this a has got multiplied by both
the terms of the second bracket now we
have to start with this minus B this
minus B will get multiplied by a first
and then it will get multiplied by this
B so minus B getting multiplied by a
will be minus a B actually it will be
minus B a but we always write variables
in alphabetical order so it will be
written as minus a B then minus B gets
multiplied by B so minus minus plus so
this term will be plus B Square now let
us solve it further
a square can be written as it is minus a
B minus a B are two negative abs that is
minus 2 a B and then we have this B
Square
you
so this was the formula for a minus B
whole square so this was our second
identity which says a minus B whole
square is equal to a square minus 2a B
plus B Square
so now let's check a third agility which
needs to be proven the third identity
was a square minus B square is equal to
a minus B multiplied by a plus B
let me write down this third identity
here it says a square minus B square is
equal to a minus B multiplied by a plus
B in this case we will start from the
right hand side so what is right hand
side the right hand side is a minus B
multiplied by a plus B now let's repeat
the procedure of multiplication first we
will multiply this a to this a and then
we multiply that a to this B so it will
be equal to ay multiplied by a is equal
to a square and a multiplied to plus B
it will be plus a B and then we have
this minus B multiplied to a and then
this minus B multiplied to this B so
minus B multiplied to a will be minus B
a and we have to write in alphabetical
order that is minus a B and minus B
multiplied to plus B will be minus B
Square now if we look at this expression
carefully we have 1 plus a B and we have
1 minus a B so positive negative same
term will get cancelled out and we will
have a square minus B Square which is
nothing but the left hand side of the
identity so we have proved this identity
these all these are very important
identities
now let me write this identity again it
says a square minus B square is equal to
a minus B multiplied by a plus B so we
have learnt three identities this was
the third one a my a square minus B
Square is equal to a minus B multiplied
by a plus B all these identities which
we have learnt they are very useful in
solving algebraic problems these were
the identities the first one then this
was the second one and this was the
third one so these are the three
identities and these come very handy
when we solve algebraic problems so I
will expect you to remember all this and
you should also see few more videos
these where I have explained these
identities using geometrical figures
they are very interesting methods and
and if you see those videos you will
remember these identities better
plus B whole square equal to a square
plus 2 a B plus B Square and identity
like a minus B whole square equal to a
square minus 2 a B plus B Square and one
more which is a square minus B Square
equal to a minus B into a plus B can be
derived so now let's talk about the
first identity which is a plus B whole
square we will start from the left hand
side and try to reach towards the
right-hand side of the identity so what
is the left-hand side left-hand side is
a plus B whole square so let's start
from here what is the meaning of a plus
B whole square a plus B whole square is
actually equal to a plus B multiplied to
itself okay
so how we multiply while multiplying two
expressions here the two expressions are
a plus B and a plus B so while
multiplying what we do we take the first
term of the first expression and then we
multiply it to the first term of the
second expression and then again we take
first term of the first expression and
then we multiply it to the second term
of the second expression and if we have
more than two terms we have to continue
this procedure okay then we start with
the second term of the first expression
and repeat the procedure we multiply the
second term of the first expression to
the first term of the second expression
and then we multiplied the second term
of the first section expression to the
second term of the second expression and
if we have more terms we have to repeat
this procedure okay so let's write what
we will get by doing this procedure it
will be equal to first a will get
multiplied to a which will be a
multiplied by a then we have a plus sign
why because this B is positive a is
getting multiplied to B so we will get a
B and now since there were only two
terms in the second expression we will
start with the second term of the first
expression that is B we will multiply
this B by the first term of the second
expression which is a B multiplied by a
is B a and we write it in alphabetical
order that will be a B and then B will
get multiplied to B that is the second
term of the first expression and the
second term of the second expression
both are B B
multiplied by B is like this now let us
solve this further a multiplied by a is
a square a B and a B there are two ABS
so plus 2 a B and then we have plus P
multiplied by B that is B Square so it
is plus B Square so what is the value of
a plus B whole square a plus B whole
square is actually equal to a square
plus 2 a B plus B Square so this was the
derivation of our first identity which
says a plus B whole square is equal to a
square plus 2 a B plus B Square ok now
let's talk about the second identity
second identity is this which says a
minus B whole square is equal to a
square minus 2 a B plus B square so what
I'll do I'll write the left-hand side of
the expression here the left-hand side
of the expression is a minus B whole
square okay so what is the meaning of a
minus B whole square a minus B whole
square is nothing but a minus B
multiplied by itself that is a minus P a
minus B whole square is nothing but a
minus B
multiplied by a minus B again we have to
repeat the procedure of multiplication
this a will get multiplied by this a
first and then it will get multiplied to
this B so this will give us a multiplied
by a is a squared this time I am writing
directly okay and then this a multiplied
by minus B so it will be minus a B why
this minus sign in multiplication if -
get multiplied by minus we will get a
positive term if plus gets multiplied by
plus then also we get a positive term
but if a minus gets multiplied by plus
or a plus gets multiplied by minus we
will get a minus term this we have
learnt in the video where I am
explaining you how to multiply two
algebraic expressions you can refer that
video okay so since this is there so
next term what we have to do we have to
since this a has got multiplied by both
the terms of the second bracket now we
have to start with this minus B this
minus B will get multiplied by a first
and then it will get multiplied by this
B so minus B getting multiplied by a
will be minus a B actually it will be
minus B a but we always write variables
in alphabetical order so it will be
written as minus a B then minus B gets
multiplied by B so minus minus plus so
this term will be plus B Square now let
us solve it further
a square can be written as it is minus a
B minus a B are two negative abs that is
minus 2 a B and then we have this B
Square
you
so this was the formula for a minus B
whole square so this was our second
identity which says a minus B whole
square is equal to a square minus 2a B
plus B Square
so now let's check a third agility which
needs to be proven the third identity
was a square minus B square is equal to
a minus B multiplied by a plus B
let me write down this third identity
here it says a square minus B square is
equal to a minus B multiplied by a plus
B in this case we will start from the
right hand side so what is right hand
side the right hand side is a minus B
multiplied by a plus B now let's repeat
the procedure of multiplication first we
will multiply this a to this a and then
we multiply that a to this B so it will
be equal to ay multiplied by a is equal
to a square and a multiplied to plus B
it will be plus a B and then we have
this minus B multiplied to a and then
this minus B multiplied to this B so
minus B multiplied to a will be minus B
a and we have to write in alphabetical
order that is minus a B and minus B
multiplied to plus B will be minus B
Square now if we look at this expression
carefully we have 1 plus a B and we have
1 minus a B so positive negative same
term will get cancelled out and we will
have a square minus B Square which is
nothing but the left hand side of the
identity so we have proved this identity
these all these are very important
identities
now let me write this identity again it
says a square minus B square is equal to
a minus B multiplied by a plus B so we
have learnt three identities this was
the third one a my a square minus B
Square is equal to a minus B multiplied
by a plus B all these identities which
we have learnt they are very useful in
solving algebraic problems these were
the identities the first one then this
was the second one and this was the
third one so these are the three
identities and these come very handy
when we solve algebraic problems so I
will expect you to remember all this and
you should also see few more videos
these where I have explained these
identities using geometrical figures
they are very interesting methods and
and if you see those videos you will
remember these identities better
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Oct 24, 2024 Last updated |
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