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Page No 107:
Question 1:
Factorize:
1.
Answer:
The given expression to be factorized is
Take common x from the first two terms and -3 from the last two terms. That is
Finally, take common x2 + 1from the two terms. That is
We cannot further factorize the expression.
So, the required factorization is.
Page No 107:
Question 2:
Factorize:
2. a(a+b)3 − 3a2b (a + b)
Answer:
The given expression to be factorized is
Take common from the two terms. That is
Expand the term within the second braces.
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 3:
Answer:
The given expression to be factorized is
We know that
The given expression then becomes
Take common from the two terms. That is
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 4:
Factorize:
a2x2 + (ax2 + 1)x + a
Answer:
The given expression to be factorized is
Simplify the middle term. That is
Take common from the first two terms and 1 from the last two terms. That is
Finally, take commonfrom the two terms. That is
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 5:
Factorize:
x2 + y − xy − x
Answer:
The given expression to be factorized is
Rearrange the given expression as
Take common x from the first two terms and -1 from the last two terms. That is
Finally, take commonfrom the two terms. That is
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 6:
Factorize:
x3 − 2x2y + 3xy2 − 6y3
Answer:
The given expression to be factorized is
Take common from the first two terms and from the last two terms. That is
Finally, take commonfrom the two terms. That is
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 7:
Factorize:
6ab − b2 + 12ac − 2bc
Answer:
The given expression to be factorized is
Take common b from the first two terms and from the last two terms. That is
Finally, take commonfrom the two terms. That is
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 8:
Factorize:
Answer:
The given expression to be factorized is
Take common 4 from the last two terms. That is
Again take commonfrom the two terms of the above expression.
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 9:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 10:
Factorize:
a2 + 2ab +b2 − c2
Answer:
The given expression to be factorized is
This can be arrange in the form
Substitutingin the above expression, we get.
Put.
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 11:
Factorize:
a2 + 4b2 − 4ab − 4c2
Answer:
The given expression to be factorized is
This can be arrange in the form
Substitute.
Put.
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 12:
Factorize:
x2 − y2 − 4xz + 4z2
Answer:
The given expression to be factorized is
Rearrange the terms as
Substitutingin the avove expression,
Put.
We cannot further factorize the expression.
So, the required factorization ofis.
Page No 107:
Question 13:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
Take common x from the first two terms andfrom the last two terms,
Finally take commonfrom the above expression,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 14:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
Take common x from the first two terms andfrom the last two terms,
Finally take commonfrom the above expression,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 15:
Factorize:
(a − b + c)2 + (b − c + a)2 + 2(a − b + c) (b − c + a)
Answer:
The given expression to be factorized is
This can be written as
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 16:
Give possible expressions for the length and breadth of the rectangle having 35y2 + 13y − 12 as its area.
Answer:
The area of the rectangle is
First we will factorize the above expression. This can be written in the form
Take commonfrom the first two terms andfrom the last two terms,
Finally take commonfrom the above expression,
The area of a rectangle having length a and breadth bis ab.
Here we don’t know the bigger or the smaller factor. So, the two possibilities are
(i) Length isand breadth is
(ii) Length is and breadth is
Page No 107:
Question 17:
What are the possible expressions for the dimensions of the cuboid whose volume is 3x2− 12x.
Answer:
The volume of the cuboid is
First we will factorize the above expression.
Take commonfrom the two terms of the above expression,
The volume of a cuboid having length, breadth b and height is.
Here the word ‘dimensions’ stands for the length, breadth and height of the cuboid. So, the three possibilities are
(i) Length is, breadth is x and height is
(ii) Length is x , breadth isand height is
(iii) Length is, breadth isand height is x
There are many other possibilities also, because we can consider the product of two simple factors as a single factor.
Page No 107:
Question 18:
Factorize:
a2 + b2 + 2bc − c2
Answer:
The given expression to be factorized is
This can be arrange in the form
Substitutingin the above expression, we get.
Put.
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 19:
Factorize:
xy9 − yx9
Answer:
The given expression to be factorized is
This can be written in the form
Take commonfrom the two terms of the above expression
We cannot further factorize the expression.
So, the required factorization of is
Page No 107:
Question 20:
Factorize:
x4 + x2y2 + y4
Answer:
The given expression to be factorized is
Add and subtract the termin the given expression.
Substitutingin the above expression, we get
Putin the above expression,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 21:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
Take common x from the first two terms andfrom the last two terms.
Finally, take commonfrom the above expression. Then we have
We cannot further factorize the expression.
So, the required factorization is.
Page No 107:
Question 22:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
Take common x from the first two terms andfrom the last two terms,
Finally take commonfrom the above expression,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 23:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
Take common x from the first two terms andfrom the last two terms. Then we have
Finally take commonfrom the above expression. Then we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 24:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
Take common x from the first two terms andfrom the last two terms. Then we have
Finally take commonfrom the above expression,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 25:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
Take common x from the first two terms andfrom the last two terms,
Finally take commonfrom the above expression,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 26:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
Take commonfrom the first two terms andfrom the last two terms,
Finally take commonfrom the above expression,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 27:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
Take commonfrom the first two terms andfrom the last two terms,
Finally take commonfrom the above expression,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 28:
Factorize:
Answer:
The given expression to be factorized is
This can be written in the form
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 29:
Factorize:
7(x − 2y)2 − 25(x − 2y) + 12
Answer:
The given expression to be factorized is
Substitutingin the above expression, we get
This can be written in the form
Take commonfrom the first two terms andfrom the last two terms,
Finally take commonfrom the above expression,
Put in the above expression,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 30:
Factorize:
2(x + y)2 − 9(x + y) − 5
Answer:
The given expression to be factorized is
Substitutingin the above expression, we get
This can be written in the form
Take commonfrom the first two terms andfrom the last two terms,
Finally take commonfrom the above expression,
Put. Then we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 31:
Factorize:
4(x − y)2 − 12(x − y) (x + y) + 9(x + y)2
Answer:
The given expression to be factorized is
Substituting andin the above expression, we get
=
This can be arrange in the form
Putand.
Take common -1 from the expression within the braces.
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 32:
Factorize:
a2 + b2 + 2(ab + bc + ca)
Answer:
The given expression to be factorized is
This can be written as
Take commonfrom the last two terms.
Finally, take common from the two terms of the above expression.
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 33:
Factorize:
(x+2) (x2+25) − 10x2 − 20x
Answer:
The given expression to be factorized is
Take common from the last two terms. That is
Again take commonfrom the two terms of the above expression. Then
We cannot further factorize the expression.
So, the required factorization of is.
Page No 107:
Question 34:
Factorize:
Answer:
The given expression to be factorized is
We have
Use the above result in the original expression to get
Substituting in the above , we get
Put.
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 1:
y3 + 125
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula for sum of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 2:
1 − 27a3
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula for difference of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 3:
8x3y3 + 27a3
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula for sum of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 4:
64a3 − b3
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula for difference of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 5:
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula for difference of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 6:
10x4y − 10xy4
Answer:
The given expression to be factorized is
Take common from the two terms,. Then we have
This can be written in the form
Recall the formula for difference of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 7:
54x6y + 2x3y4
Answer:
The given expression to be factorized is
Take common from the two terms,. Then we have
This can be written in the form
Recall the formula for sum of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 8:
32a3 + 108b3
Answer:
The given expression to be factorized is
Take common from the two terms,. Then we have
This can be written in the form
Recall the formula for sum of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 9:
(a − 2b)3 − 512b3
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula for difference of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 10:
8x2y3 − x5
Answer:
The given expression to be factorized is
Take common. Then we have
This can be written as
Recall the formula for difference of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 11:
1029 − 3x3
Answer:
The given expression to be factorized is
Take common 3. Then we have from the above expression,
This can be written as
Recall the formula for difference of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 12:
x3y3 + 1
Answer:
The given expression to be factorized is
This can be written as
Recall the formula for sum of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 13:
x4y4 − xy
Answer:
The given expression to be factorized is
Take common. Then we have from the above expression,
This can be written as
Recall the formula for difference of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 14:
a3 + b3 + a + b
Answer:
The given expression to be factorized is
This can be written as
=
Recall the formula for sum of two cubes
Using the above formula, we have
Take common. Then we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 15:
Simplify:
(i)
(ii)
(iii)
Answer:
(i) The given expression is
Assumeand. Then the given expression can be rewritten as
Recall the formula for sum of two cubes
Using the above formula, the expression becomes
Note that both and b are positive. So, neithernor any factor of it can be zero.
Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes
(ii) The given expression is
Assumeand. Then the given expression can be rewritten as
Recall the formula for difference of two cubes
Using the above formula, the expression becomes
Note that both, b is positive and unequal. So, neithernor any factor of it can be zero.
Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes
(iii) The given expression is
Assumeand. Then the given expression can be rewritten as
Recall the formula for difference of two cubes
Using the above formula, the expression becomes
Note that both, b is positive and unequal. So, neithernor any factor of it can be zero.
Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes
Page No 111:
Question 16:
(a + b)3 − 8(a − b)3
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula for difference of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 17:
(x + 2)3 + (x − 2)3
Answer:
The given expression to be factorized is
Recall the formula for sum of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 18:
a12+ b12
Answer:
The given expression to be factorized is
This can be written as
Recall the formula for difference of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 19:
x3 + 6x2 + 12x + 16
Answer:
The given expression to be factorized is
This can be written as
Take common x2 from first two terms, 2x from the next two terms andfrom the last two terms. Then we have,
Finally, take common. Then we get,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 20:
Answer:
The given expression to be factorized is
This can be written as
Recall the formula for sum of two cubes
Using the above formula and taking common from the last two terms, we get
Take common. Then we have,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 21:
a3 + 3a2b + 3ab2 + b3 − 8
Answer:
The given expression to be factorized is
Recall the well known formula
The given expression can be written as
Recall the formula for difference of two cubes
Using the above formula and taking common –2 from the last two terms, we get
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 22:
8a3 − b3 − 4ax + 2bx
Answer:
The given expression to be factorized is
The given expression can be written as
Recall the formula for difference of two cubes
Using the above formula and taking common from the last two terms, we get
Take common. Then we have,
We cannot further factorize the expression.
So, the required factorization of is.
Page No 111:
Question 23:
Simplify: (2x – 5y)3 – (2x + 5y)3
Answer:
Ans
Page No 114:
Question 1:
Factorize:
64a3 + 125b3 + 240a2b + 300ab2
Answer:
The given expression to be factorized is
This can be written in the form
Take common from the last two terms,
This can be written in the following form
Recall the formula for the cube of the sum of two numbers
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 114:
Question 2:
125x3 − 27y3 − 225x2y + 135xy2
Answer:
The given expression to be factorized is
This can be written in the form
Take common from the last two terms,. Then we get
This can be written in the following form
Recall the formula for the cube of the difference of two numbers
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is.
Page No 114:
Question 3:
Answer:
The given expression to be factorized is
This can be written in the form
Take common from the last two terms,. Then we get
This can be written in the following form
Recall the formula for the cube of the sum of two numbers
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is.
Page No 114:
Question 4:
8x3 + 27y3 + 36x2y + 54xy2
Answer:
The given expression to be factorized is
This can be written in the form
Take common from the last two terms. Then we get
This can be written in the following form
Recall the formula for the cube of the sum of two numbers
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is.
Page No 114:
Question 5:
a3 − 3a2b + 3ab2 − b3 + 8
Answer:
The given expression to be factorized is
This can be written in the form
Take common from the third and fourth terms. Then we get
This can be written in the following form
Recall the formula for the cube of the difference of two numbers
Using the above formula, we have
This can be written in the following form
Recall the formula for the sum of two cubes
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is ofis.
Page No 114:
Question 6:
x3 + 8y3 + 6x2y + 12xy2
Answer:
The given expression to be factorized is
This can be written in the form
Take common from the last two terms. Then we get
This can be written in the following form
Recall the formula for the cube of the sum of two numbers
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is.
Page No 114:
Question 7:
8x3 + y3 + 12x2y + 6xy2
Answer:
The given expression to be factorized is
This can be written in the form
Take common 6xy from the last two terms,. Then we get
This can be written in the following form
Recall the formula for the cube of the sum of two numbers
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is.
Page No 114:
Question 8:
8a3 + 27b3 + 36a2b + 54ab2
Answer:
The given expression to be factorized is
This can be written in the form
Take common from the last two terms,. Then we get
This can be written in the following form
Recall the formula for the cube of the sum of two numbers
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is.
Page No 114:
Question 9:
8a3 − 27b3 − 36a2b + 54ab2
Answer:
The given expression to be factorized is
This can be written in the form
Take common – 18ab from the last two terms,. Then we get
This can be written in the following form
Recall the formula for the cube of the difference of two numbers
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is ofis.
Page No 114:
Question 10:
x3 − 12x(x − 4) − 64
Answer:
The given expression to be factorized is
This can be written in the form
Take common – 12x from the last two terms,. Then we get
This can be written in the following form
Recall the formula for the cube of the difference of two numbers
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is.
Page No 114:
Question 11:
a3x3 − 3a2bx2 + 3ab2x − b3
Answer:
The given expression to be factorized is
This can be written in the form
Take common from the last two terms,. Then we get
This can be written in the following form
Recall the formula for the cube of the difference of two numbers
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is.
Page No 119:
Question 1:
Factorize each of the following expressions:
a3 + 8b3 + 64c3 − 24abc
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization of is.
Page No 119:
Question 2:
x3 − 8y3 + 27z3 + 18xyz
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is ofis.
Page No 119:
Question 3:
27x3 − y3 − z3 − 9xyz
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is .
Page No 119:
Question 4:
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is ofis .
Page No 119:
Question 5:
8x3 +27y3 − 216z3 + 108xyz
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is ofis .
Page No 119:
Question 6:
125 + 8x3 − 27y3 + 90xy
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is ofis .
Page No 119:
Question 7:
8x3 − 125y3 + 180xy + 216
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is ofis .
Page No 119:
Question 8:
Multiply:
(i) x2 + y2 + z2 − xy + xz + yz by x + y − z
(ii) x2 + 4y2 + z2 + 2xy + xz − 2yz by x − 2y − z
(iii) x2 + 4y2 + 2xy − 3x + 6y + 9 by x − 2y + 3
(iv) 9x2 + 25y2 + 15xy + 12x − 20y + 16 by 3x − 5y + 4
(v) x2 + 4y2 + z2 + 2xy + xz – 2yz by (−z + x – 2y)
(vi) 4x2 + y2 + 9z2 + 2xy + 3yz – 6xz by (2x – y + 3z)
Answer:
(i) The given expression is .
We have to multiply the above expression by.
The required product is
Recall the formula
Using the above formula, we have
(ii) The given expression is .
We have to multiply the above expression by.
The required product is
Recall the formula
Using the above formula, we have
(iii) The given expression is .
We have to multiply the above expression by.
The required product is
Recall the formula
Using the above formula, we have
(iv) The given expression is .
We have to multiply the above expression by.
The required product is
Recall the formula
Using the above formula, we have
(v) The given expression is .
We have to multiply the above expression by .
The required product is
Recall the formula
Using the above formula, we have
(vi) The given expression is 4x2 + y2 + 9z2 + 2xy + 3yz – 6xz.
We have to multiply the above expression by (2x – y + 3z).
The required product is
(2x – y + 3z)(4x2 + y2 + 9z2 + 2xy + 3yz – 6xz)
= {2x + (– y) + 3z}{(2x)2 + (– y)2 + (3z)2 – (2x)(–y) – (–y)(3z) – (3z)(2x)}
Recall the formula
Using the above formula, we have
= (2x)3 + (–y)3 + (3z)3 – 3(2x)(–y)(3z)
= 8x3 – y3 + 27z3 + 18xyz
Page No 119:
Question 9:
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is .
Page No 119:
Question 10:
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is.
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Question 11:
Answer:
The given expression to be factorized is
This can be written in the form
Recall the formula
Using the above formula, we have
We cannot further factorize the expression.
So, the required factorization is of is .
Page No 119:
Question 12:
Without actually computing the cubes, find the value of 483 – 303 – 183.
Answer:
Let x = 48, y = −30, z = −18
We know,
if x + y + z = 0
then, x3 + y3 + z3 = 3xyz
Page No 119:
Question 13:
(3x − 2y)3 + (2y − 4z)3 + (4z − 3x)3
Answer:
The given expression to be factorized is
Let, and. Then the given expression becomes
Note that
Recall the formula
When, this becomes
So, we have the new formula
,
when.
Using the above formula, the given expression can be written as
Put, and. Then we have
We cannot further factorize the expression.
So, the required factorization is ofis.
Page No 119:
Question 14:
(2x − 3y)3 + (4z − 2x)3 + (3y − 4z)3
Answer:
The given expression to be factorized is
Let, and. Then the given expression becomes
Note that
Recall the formula
When, this becomes
So, we have the new formula
, when.
Using the above formula, the given expression can be written as
Put, and. Then we have
We cannot further factorize the expression.
So, the required factorization is ofis .
Page No 119:
Question 15:
(a − 3b)3 + (3b − c)3 + (c − a)3
Answer:
The given expression to be factorized is
Let, and. Then the given expression becomes
Note that
Recall the formula
When, this becomes
So, we have the new formula
, when.
Using the above formula, the given expression can be written as
Put, and. Then we have
We cannot further factorize the expression.
So, the required factorization is ofis.
Page No 119:
Question 16:
(x – 2y)3 + (2y – 3z)3 + (3z – x)3
Answer:
Page No 119:
Question 17:
Answer:
The given expression to be factorized is
Let, and. Then the given expression becomes
Note that
Recall the formula
When, this becomes
So, we have the new formula
, when.
Using the above formula, the given expression can be written as
Put, and.
Then we have
We cannot further factorize the expression.
So, the required factorization is of is
Page No 119:
Question 18:
Find the value of x3 + y3 − 12xy + 64, when x + y =−4
Answer:
The given expression is
It is given that
The given expression can be written in the form
Recall the formula
Using the above formula, we have
Page No 119:
Question 19:
If a, b, c are all non-zero and a + b + c = 0, prove that .
Answer:
Page No 120:
Question 1:
If a + b + c = 0, then write the value of a3 + b3 + c3.
Answer:
Recall the formula
When, we have
Page No 120:
Question 2:
If a2 + b2 + c2 = 20 and a + b + c = 0, find ab + bc + ca.
Answer:
Recall the formula
Given that
Then we have
Page No 120:
Question 3:
If a + b + c = 9 and ab + bc + ca = 40, find a2 + b2 +c2.
Answer:
Recall the formula
Given that
,
Then we have
Page No 120:
Question 4:
If a2 + b2 + c2 = 250 and ab + bc + ca = 3, find a + b + c.
Answer:
Recall the formula
Given that
,
Then we have
Page No 120:
Question 5:
Write the value of 253 − 753 + 503.
Answer:
The given expression is
Let, and. Then the given expression becomes
Note that
Recall the formula
When, this becomes
So, we have the new formula
, when.
Using the above formula, the value of the given expression is
Page No 120:
Question 6:
Write the value of 483 − 303 − 183.
Answer:
The given expression is
Let, and. Then the given expression becomes
Note that
Recall the formula
When, this becomes
So, we have the new formula
, when.
Using the above formula, the value of the given expression is
Page No 120:
Question 7:
Write the value of
Answer:
The given expression is
Let, and. Then the given expression becomes
Note that:
Recall the formula
When, this becomes
So, we have the new formula
, when.
Using the above formula, the value of the given expression is
Page No 120:
Question 8:
Write the value of 303 + 203 − 503.
Answer:
The given expression is
Let, and. Then the given expression becomes
Note that
Recall the formula
When, this becomes
So, we have the new formula
, when.
Using the above formula, the value of the given expression is
Page No 120:
Question 9:
Factorize: x4 + x2 + 25.
Answer:
The given expression to be factorized is
This can be written in the form
We cannot further factorize the expression.
So, the required factorization is.
Page No 120:
Question 10:
Factorize : x2 − 1 − 2a − a2
Answer:
The given expression to be factorized is
Take commonfrom the last three terms and then we have
We cannot further factorize the expression.
So, the required factorization is.
Page No 121:
Question 1:
The factorized form of the expression y2 + (x – 1)y – x is ____________.
Answer:
Hence, the factorized form of the expression y2 + (x – 1)y – x is (y ​– 1)(y + x).
Page No 121:
Question 2:
The factorized form of a3 + (b – a)3 – b3 is ____________.
Answer:
Hence, the factorized form of a3 + (b – a)3 – b3 is 3ab(a – b).
Page No 121:
Question 3:
If
Answer:
Hence, if
Page No 121:
Question 4:
Factorization of the polynomial gives ____________.
Answer:
Hence, factorization of the polynomial gives .
Page No 121:
Question 5:
The polynomial x2 + y2 – z2 – 2xy on factorization gives _____________.
Answer:
Hence, the polynomial x2 + y2 – z2 – 2xy on factorization gives .
Page No 121:
Question 6:
The factors of the expression are ____________.
Answer:
Hence, the factors of the expression are .
Page No 121:
Question 7:
The polynomial , x6 + 64y6 on factorization gives _____________.
Answer:
Hence, the polynomial x6 + 64y6 on factorization gives .
Page No 121:
Question 8:
The factorization form of a4 + b4 – a2b2 is _____________.
Answer:
Hence, the factorization form of a4 + b4 – a2b2 is .
Page No 121:
Question 9:
If , then the value of is ____________.
Answer:
Hence, the value of is 1090.
Page No 121:
Question 10:
The factorized form of is _____________.
Answer:
Hence, the factorized form of is
Page No 121:
Question 11:
The factorized form of a3 + b3 + 3ab – 1 is ____________.
Answer:
Hence, the factorized form of a3 + b3 + 3ab – 1 is
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