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Factorization
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Factorisation of Polynomials using Algebraic Identities
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The factors of
81
p
2
-
144
q
2
are
(a)
(
3
p
-
4
q
)
,
(
3
p
+
4
q
)
(b)
9
,
(
3
p
-
4
q
)
,
(
3
p
+
4
q
)
(c)
9
,
3
p
2
-
4
q
2
(d)
9
,
3
p
2
+
4
q
2
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answered this correctly
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Important Questions on Factorization
MEDIUM
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Let
x
=
50
+
7
1
3
-
50
-
7
1
3
. Then-
MEDIUM
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Let
a
,
b
,
c
be non-zero real numbers such that
a
+
b
+
c
=
0
;
let
q
=
a
2
+
b
2
+
c
2
&
r
=
a
4
+
b
4
+
c
4
. Then
HARD
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Consider the equation
1
+
a
+
b
2
=
3
1
+
a
2
+
b
2
,
where
a
,
b
are real numbers. Then
HARD
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Each of the numbers
x
1
,
x
2
,
…
,
x
101
is
±
1
.
What is the smallest positive value of
∑
1
≤
i
≤
j
≤
101
x
i
x
j
?
MEDIUM
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Factorise
a
2
b
+
c
+
b
2
c
+
a
+
c
2
a
+
b
+
3
a
b
c
.
HARD
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
If
a
,
b
,
c
≥
4
are integers, not all equal and
4
a
b
c
=
a
+
3
b
+
3
c
+
3
, then what is the value of
a
+
b
+
c
?
MEDIUM
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
If
a
,
b
,
c
are real numbers such that
a
+
b
+
c
=
0
and
a
2
+
b
2
+
c
2
=
1
, then
3
a
+
5
b
-
8
c
2
+
-
8
a
+
3
b
+
5
c
2
+
5
a
-
8
b
+
3
c
2
is equal to
HARD
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Let
S
=
a
2
+
b
2
+
c
2
a
b
+
b
c
+
c
a
:
a
,
b
,
c
∈
R
,
a
b
+
b
c
+
c
a
≠
0
where
R
is the set of real numbers. Then,
S
equals
MEDIUM
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Find the factors of the Polynomial
3
x
2
-
2
x
-
1
.
MEDIUM
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Let
x
,
y
,
z
be non-zero real numbers such that
x
y
+
y
z
+
z
x
=
7
and
y
x
+
z
y
+
x
z
=
9
,
then
x
3
y
3
+
y
3
z
3
+
z
3
x
3
-
3
is equal to
MEDIUM
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Integers
a
,
b
,
c
satisfy
a
+
b
-
c
=
1
and
a
2
+
b
2
-
c
2
=
-
1
.
What is the sum of all possible values of
a
2
+
b
2
+
c
2
?
HARD
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Find the largest value of
a
b
such that the positive integers
a
,
b
>
1
satisfy
a
b
b
a
+
a
b
+
b
a
=
5329
.
HARD
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Find the number of positive integers less than
101
that cannot be written as the difference of two squares of integers.
MEDIUM
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
If
x
4
-
83
x
2
+
1
=
0
, then a value of
x
3
-
x
-
3
can be:
EASY
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
a
,
b
,
c
are positive real numbers such that
a
2
+
b
2
=
c
2
and
a
b
=
c
.
Determine the value of
(
a
+
b
+
c
)
(
a
+
b
-
c
)
(
b
+
c
-
a
)
(
c
+
a
-
b
)
c
2
.
HARD
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Let
x
and
y
be two
2
-digit numbers such that
y
is obtained by reversing the digits of
x
.
Suppose they also satisfy
x
2
-
y
2
=
m
2
for some positive integer
m
.
The value of
x
+
y
+
m
is.
MEDIUM
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
The product of two consecutive natural numbers which are multiples of
3
is equal to
810
. Find the two numbers.
HARD
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
What is the smallest prime number
p
such that
p
3
+
4
p
2
+
4
p
has exactly
30
positive divisors?
EASY
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Factorise:
2
a
+
6
b
–
3
(
a
+
3
b
)
2
.
EASY
Mathematics
>
Algebra
>
Factorization
>
Factorisation of Polynomials using Algebraic Identities
Factorise:
x
(
x
+
3
)
+
5
(
x
+
3
)
