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1. Factorize each of the following expression
(i) – 5a^2 + 5ab – 52a
(ii) x^2yz + xy^2z + xyz^2
(iii) – 4x^5 – 16x^3y – 20x^2y^2
(iv) x^3yz + 4xy^3 + 14x^3
(v) – a^2 + 3a + a – p – 2
(vi) x^2p + y^2p + z^2p – x^2q – y^2q – z^2q
Solution:
(i) The given expression is – 5a^2 + 5ab – 52a.
Factor out the common term from the above expression.That is,
a(- 5a + 5b – 52).
– 5a^2 + 5ab – 52a is equal to a(- 5a + 5b – 52).
(ii) The given expression is x^2yz + xy^2z + xyz^2
Factor out the common term from the above expression.That is,
xyz(x + y + z).
x^2yz + xy^2z + xyz^2 is equal to xyz(x + y + z).
(iii) The given expression is – 4x^5 – 16x^3y – 20x^2y^2.
Factor out the common term from the above expression. That is,
– 4x^2(x^3 + 4xy + 5y^2).
– 4x^5 – 16x^3y – 20x^2y^2 is equal to 4x^2(x^3 + 4xy + 5y^2).
(iv) The given expression is x^3yz + 4xy^3 + 14x^3
Factor out the common term from the above expression.That is,
x(x^2yz + 4y^2 + 14x^2).
x^3yz + 4xy^3 + 14x^3is equal to x(x^2yz + 4y^2 + 14x^2).
(v) The given expression is – a^2 + 3a + a – p – 2.
Factor out the common term from the above expression.That is,
a(-a + 3 + 1) – (p + 2).
(a – 1) (p – a + 2).
– a^2 + 3a + a – p – 2 is equal to (a – 1) (p – a + 2).
(vi) The given expression is x^2p + y^2p + z^2p – x^2q – y^2q – z^2q.
Factor out the common term from the above expression.That is,
x^2(p – q) + y^2(p – q) + z^2(p – q) = (p – q) (x^2 + y^2 + z^2).
x^2p + y^2p + z^2p – x^2q – y^2q – z^2q is equal to (p – q) (x^2 + y^2 + z^2).
2. Factorize using the formula of the difference of two squares
(i) 25a^2 – 36a^2b^2
(ii) 49x^2 – y^4
(iii) 81x^4 – y^2
(iv) a^3 – 25a.
(v) 32x^2y – 200y^3
(vi) 4x^2 + 12xy + 9y^2 – 9.
Solution:
(i) The given expression is 25a^2 – 36a^2b^2.
We can write it as (5a)^2 – (6ab)^2.
Factor out the common term from the above expression. That is,
a^2[(5)^2 – (6b)^2].
By comparing the above expression, it matches with the basic expression (a)^2 – (b)^2 = (a + b) (a – b).
So, a^2[(5)^2 – (6b)^2] = a^2(5 + 6b) (5 – 6b).
25a^2 – 36a^2b^2 is equal to a^2(5 + 6b) (5 – 6b).
(ii) The given expression is 49x^2 – y^4.
We can write it as (7x)^2 – (y^2)^2.
By comparing the above expression, it matches with the basic expression (a)^2 – (b)^2 = (a + b) (a – b).
So, (7x)^2 – (y^2)^2= (7x + y^2) (7x – y^2).
49x^2 – y^4 is equal to (7x + y^2) (7x – y^2).
(iii) The given expression is 81x^4 – y^2.
We can write it as (9x^2)^2 – (y)^2.
By comparing the above expression, it matches with the basic expression (a)^2 – (b)^2 = (a + b) (a – b).
So, (9x^2)^2 – (y)^2 = (9x^2 + y) (9x^2 – y).
81x^4 – y^2 is equal to (9x^2 + y) (9x^2 – y).
(iv) The given expression is a^3 – 25a
factor out the common terms from the above expression. That is,
a (a^2 – 25).
We can write it as a (a^2 – 5^2).
By comparing the above expression, it matches with the expression a^2 – b^2 = (a + b) (a – b).
So, a (a^2 – 5^2) = a (a + 5) (a – 5).
a^3 – 25a is equal to a(a + 5) (a – 5).
(v) The given expression is 32x^2y – 200y^3.
Factor out the common terms from the above expression. That is,
8y (4x^2 – 25y^2).
We can write it as 8y ((2x)^2 – (5y)^2).
By comparing the above expression, it matches the expression a^2 – b^2 = (a + b) (a – b).
So,8y ((2x)^2 – (5y)^2) = 8y (2x + 5y) (2x – 5y).
32x^2y – 200y^3 is equal to 8y(2x + 5y) (2x – 5y).
(vi) The given expression is 4x^2 + 12xy + 9y^2 – 9.
We can write it as (2x)^2 + 2(2x) (3y) + (3y)^2 – 9.
By comparing the above expression. It matches with the expression a^2 + 2ab + b^2 = (a + b)^2.
Then, (2x)^2 + 2(2x) (3y) + (3y)^2 – 9 = (2x + 3y)^2 – 9.
(2x + 3y)^2 – 9 = (2x + 3y)^2 – (3)^2
By comparing the above expression, it matches the expression a^2 – b^2 = (a + b) (a – b).
So, (2x + 3y)^2 – (3)^2 = (2x + 3y + 3) (2x + 3y – 3).
4x^2 + 12xy + 9y^2 – 9 is equal to (2x + 3y + 3) (2x + 3y – 3).
3. Find the values of
(i) (8)^2 – (6)^2
(ii) (27^2/3)^2 – (8 1/3)^2
(iii) (42)^2 – (14)^2
(iv) (10003)^2 – (9997)^2
(v) (9.2)^2 – (0.8)^2
Solution:
(i) The given expression is (8)^2 – (6)^2.
By comparing the above equation, it matches with the basic expression
(a)^2 – (b)^2 = (a + b) (a – b).
(8)^2 – (6)^2 = (8 + 6) (8 – 6).
= (14) (2) = 28.
(8)^2 – (6)^2 = 28.
(ii) The given expression is (27^2 / 3)^2 – (8 ^1/3)^2.
We can write it as
(3^3 * 2 / 3)^2 – (2^3 * 1 / 3)^2 = (3^2)^2 – (2)^2.
By comparing the above equation, it matches with the basic expression
(a)^2 – (b)^2 = (a + b) (a – b).
(9)^2 – (2)^2 = (9 + 2) (9 – 2) = (11) (7)= 77.
(27^2 / 3)^2 – (8 ^1/3)^2 is equal to 77
(iii) The given expression is (42)^2 – (14)^2.
By comparing the above expression with the basic expressions, it matches with the expression a^2 – b^2 = (a + b) (a – b).
(42)^2 – (14)^2 = (42 + 14) (42 – 14) = (56) (28) = 1568.
(42)^2 – (14)^2 is equal to 1568.
(iv) The given expression is (10003)^2 – (9997)^2.
By comparing the above expression with the basic expressions, it matches with the expression a^2 – b^2 = (a + b) (a – b).
(10003)^2 – (9997)^2 = (10003 + 9997) (10003 – 9997).
= (20,000) (6) = 1,20,000.
(10003)^2 – (9997)^2 is equal to 1,20,000.
(v) The given expression is (9.2)^2 – (0.8)^2.
By comparing the above expression with the basic expressions, it matches the expression a^2 – b^2 = (a + b) (a – b).
(9.2)^2 – (0.8)^2 = (9.2 + 0.8) (9.2 – 0.8) = (10) (8.4) = 84.
(9.2)^2 – (0.8)^2 = 84.
4. Factorization of trinomial
(i) 10x^2y^2 – 11xy + 3
(ii) 12a^2 + 11a – 5
(iii) 15x^2y^2 – 21xy + 6
(iv) a^4 – 10a^2 + 9
Solution:
(i) The given expression is 10x^2y^2 – 11xy + 3.
If xy = a, then 10a^2 – 11a + 3.
By comparing the above expression, it matches with the expression ax^2 + bx + c.
Here, a = 10, b = – 11, c = 3
a * c = 10 * 3 = 30, b = – 11.
30 = – 6 * ( – 5) and – 11 = – 6 + ( – 5).
So, 10a^2 – 11a + 3 = 10a^2 – 5a – 6a + 3.
10a^2 – 5a – 6a + 3 = 5a(2a – 1) – 3(2a – 1).
Factor out the common terms from the above expression. That is,
(2a – 1) (5a – 3).
Here, replace a with the xy, then
(2xy – 1) (5xy – 3).
10x^2y^2 – 11xy + 3 is equal to (2xy – 1) (5xy – 3).
(ii) The given expression is 12a^2 + 11a – 5.
By comparing the above expression, it matches with the expression ax^2 + bx + c.
Here, a = 12, b = 11, c = – 5.
a * c = 12 * (- 5) = – 60, b = 11.
-60 = 15 * ( – 4) and 11 = 15 + ( – 4).
So, 12a^2 + 11a – 5= 12a^2 – 4a + 15a – 5.
= 4a(3a – 1) + 5(3a – 1).
Factor out the common terms from the above expression. That is,
(3a – 1) (4a + 5).
12a^2 + 11a – 5 is equal to (3a – 1) (4a + 5).
(iii) The given expression is 15x^2y^2 – 21xy + 6.
If xy = a, then 15a^2 – 21a + 6.
By comparing the above expression, it matches with the expression ax^2 + bx + c.
Here, a = 15, b = – 21, c = 6.
a * c = 15 * 6 = 90 and b = – 21.
90 = – 15 * ( – 6) and – 21 = – 15 + (- 6).
15a^2 – 21a + 6 = 15a^2 – 15a – 6a + 6.
= 15a(a – 1) – 6(a – 1).
Factor out the common terms from the above expression. That is,
(a – 1) (15a – 6).
Replace a with the xy. That is,
(xy – 1) (15xy – 6).
15x^2y^2 – 21xy + 6 is equal to (xy – 1) (15xy – 6).
(iv) The given expression is a^4 – 10a^2 + 9.
We can write it as (a^2)^2 – 10a^2 + 9.
If a^2 = x, then x^2 – 10x + 9.
By comparing the above expression with the basic expressions, it matches with the ax^2 + bx + c.
Here, a = 1, b = – 10, c = 9.
a * c = 1 * 9 = 9 and b = – 10.
9 = – 9 * ( – 1) and – 10 = – 9 – 1.
x^2 – 9x – x + 9 = x(x – 9) – (x – 9).
Factoring out common terms from the above expression. That is,
(x – 9) (x – 1).
Replace the x with a^2. That is,
(a^2 – 9) (a^2 – 1).
a^4 – 10a^2 + 9 is equal to (a^2 – 9) (a^2 – 1).