For a better practice and to get full of knowledge on Factoring Polynomials Using Algebraic Identities, use Worksheet on Factoring Identities. To grade up your preparation level, we are providing a number of examples on factoring identities topics. So, practice the below worksheet on Factoring Polynomials using Algebraic Identities and develop your skills on the concept. Also, check different problems on Factorization Worksheets and prepare better for the exam.
The basic expressions for the factorization of the identities are mentioned below. They are
(i) (x + y)^2 = x^2 + y^2 + 2xy.
(ii) (x – y)^2 = x^2 + y^2 – 2xy.
(iii) x^2 – y^2 = (x + y) (x – y).
Solved Examples for Factoring Identities
1. Factor the given expressions using identity
(i) x^2 + 8x + 16.
(ii) 4a^2 – 4a + 1.
(iii) x^4 + 9y^4 + 6x^2y^2.
(iv) (x^4 – 8x^2y^2 + 16y^4) – 18.
(v) 256 – x^2 – 2xy – y^2.
Solution:
(i) The given expression is x^2 + 8x + 16.
Now, expand the given expression. That is,
x^2 + 8x + 16 = x^2 + 2(4x) + (4)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.
Here, x = x and y = 4.
So, (x + y)^2 = (x + 4)^2.
Then, x^2 + 8x + 16 is equal to (x + 4)^2.
(ii) The given expression is 4a^2 – 4a + 1.
Now, expand the given expression. That is,
4a^2 – 4a + 1 = (2a)^2 – 2(2a)(1) + (1)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 + y^2 – 2xy.
Here, x = 2a and y = 1.
So, (x – y)^2 = (2a – 1)^2.
Then, 4a^2 – 4a + 1 is equal to (2a – 1)^2.
(iii) The given expression is x^4 + 9y^4 + 6x^2y^2.
Now, expand the given expression. That is,
x^4 + 9y^4 + 6x^2y^2 = (x^2)^2 + 2(x^2)(3y^2) + (3y^2)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.
Here, x = x^2 and y = 3y^2.
So, (x + y)^2 = (x^2 + 3y^2)^2.
Then, x^4 + 9y^4 + 6x^2y^2 is equal to (x^2 + 3y^2)^2.
(iv) The given expression is (x^4 – 8x^2y^2 + 16y^4) – 18.
Now, expand the given expression. That is,
(x^4 – 8x^2y^2 + 16y^4) – 18 = [(x^2)^2 – 2(x^2) (4y^2) + (4y^2)^2] – 18.
By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 + y^2 – 2xy.
Here, x = x^2 and y = 4y^2.
So, (x – y)^2 = (x^2 – 4y^2)^2.
Then, (x^4 – 8x^2y^2 + 16y^4) – 18 is equal to (x^2 – 4y^2)^2 – 18.
(v) The given expression is 256 – x^2 – 2xy – y^2.
Now, expand the given expression. That is,
256 – x^2 – 2xy – y^2 = (16)^2 – (x^2 + 2xy + y^2)
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.
Here, x = x and y = y.
So, (x + y)^2 = (x + y)^2.
Then, 256 – x^2 – 2xy – y^2 is equal to (16)^2 – (x + y)^2.
2. Factorize the expressions
(i) 4a^2 – 12ab + 9b^2.
(ii) 36x^2 – 84xy + 49y^2.
(iii) 9x^2 + 42xy + 49y^2.
(iv) (3x – 5y)² + 2 (3x – 5y ) (2y – x) + (2y – x)²
(v) 36a² + 36a + 9
(vi) 4x4 – y4
Solution:
(i) The given expression is 4a^2 – 12ab + 9b^2.
Now, expand the given expression. That is,
4a^2 – 12ab + 9b^2 = (2a)^2 – 2(2a)(3b) + (3b)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 + y^2 – 2xy.
Here, x = 2a and y = 3b.
So, (x – y)^2 = (2a – 3b)^2.
Then, 4a^2 – 12ab + 9b^2 is equal to (2a – 3b)^2.
(ii) The given expression is 36a^2 – 84ab + 49b^2.
Now, expand the given expression. That is,
36a^2 – 84ab + 49b^2 = (6a)^2 – 2(6a)(7b) + (7b)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 + y^2 – 2xy.
Here, x = 6a and y = 7b.
So, (x – y)^2 = (6a – 7b)^2.
Then, 36a^2 – 84ab + 49b^2 is equal to (6a – 7b)^2.
(iii) The given expression is 9x^2 + 42xy + 49y^2.
Now, expand the given expression. That is,
9x^2 + 42xy + 49y^2 = (3x)^2 + 2(3x)(7y) + (7y)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.
Here, x = 3x and y = 7y.
So, (x + y)^2 = (3x + 7y)^2.
Then, 9x^2 + 42xy + 49y^2 is equal to (3x+ 7y)^2.
(iv) The given expression is (3x – 5y)^2 + 2 (3x – 5y) (2y – x) + (2y – x)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.
Here, x = (3x – 5y) and y = (2y – x).
So, (x + y)^2 = [(3x – 5y) + (2y – x)]^2.
Then, (3x – 5y)^2 + 2 (3x – 5y) (2y – x) + (2y – x)^2 is equal to [(3x – 5y) + (2y – x)]^2.
(v) The given expression is 36a² + 36a + 9.
Now, expand the given expression. That is,
36a² + 36a + 9 = (6a)^2 + 2(6a)(3) + (3)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + y^2 + 2xy.
Here, x = 6a and y = 3.
So, (x + y)^2 = (6a + 3)^2.
Then, 36a2 + 36a + 9 is equal to (6a + 3)^2.
(vi) The given expression is
4x^4 – y^4.
Now, expand the given expression. That is,
4x^4 – y^4 = (2x)^2 – (y^2)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x)^2 – (y)^2 = (x + y) (x – y).
Here, x = 2x and y = y^2.
So, (x)^2 – (y)^2= (2x + y^2) (2x – y^2).
Then, 4x^4 – y^4 is equal to (2x + y^2) (2x – y^2).
3. Factor the identities
(i) 4a² + 12ab + 9b²
(ii) a² + 22a + 121.
(iii) 9a² – 24ab + 16b²
(iv) 36a² – 36a + 9.
(v) 16a4 – 72a2b2 + 81b4
(vi) (x² + z² + 2xz) – y²
Solution:
(i) The given expression is 4a² + 12ab + 9b²
Now, expand the given expression. That is,
4a² + 12ab + 9b²= (2a)^2 + 2(2a)(3b) + (3b)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + 2xy + y^2.
Here, x = 2a and y = 3b.
So, (x + y)^2= (2a + 3b)^2.
Then, 4a2 + 12ab + 9b2 is equal to (2a + 3b)^2.
(ii) The given expression is a^2 + 22a + 121.
Now, expand the given expression. That is,
a^2 + 22a + 121= (a)^2 + 2(a)(11) + (11)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + 2xy + y^2.
Here, x = a and y = 11.
So, (x + y)^2= (a + 11)^2.
Then, a² + 22a + 121 is equal to (a + 11)^2.
(iii) The given expression is 9a² – 24ab + 16b²
Now, expand the given expression. That is,
9a² – 24ab + 16b²= (3a)^2 – 2(3a)(4b) + (4b)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 – 2xy + y^2.
Here, x = 3a and y = 4b.
So, (x – y)^2= (3a – 4b)^2.
Then, 9a² – 24ab + 16b² is equal to (3a – 4b)^2.
(iv) The given expression is 36a² – 36a + 9..
Now, expand the given expression. That is,
36a² – 36a + 9 = (6a)^2 – 2(6a)(3) + (3)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 – 2xy + y^2.
Here, x = 6a and y = 3.
So, (x – y)^2= (6a – 3)^2.
Then, 36a² – 36a + 9 is equal to (6a – 3)^2.
(v) The given expression is 16a^4 – 72a^2b^2 + 81b^4.
Now, expand the given expression. That is,
16a^4 – 72a^2b^2 + 81b^4= (4a^2)^2 – 2(4a^2)(9b^2) + (9b^2)^2.
By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 – 2xy + y^2.
Here, x = 4a^2 and y = 9b^2.
So, (x – y)^2= (4a^2 – 9b^2)^2.
Then, 16a^4 – 72a^2b^2 + 81b^4 is equal to (4a^2 – 9b^2)^2.
(vi) The given expression is (x^2 + z^2 + 2xz) – y^2.
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + 2xy + y^2.
Here, x = x and y = z.
So, (x + y)^2= (x + z)^2.
Now, (x^2 + z^2 + 2xz) – y^2 = (x + z)^2 – y^2.
By comparing the above expression, it matches the expression x^2 – y^2 = (x + y) (x – y).
So, (x + z)^2 – y^2 = (x + z + y) (x + z – y).
Then, (x^2 + z^2 + 2xz) – y^2 is equal to (x + z + y) (x + z – y).
4. Factor completely using the formula
(i) 100 – [121a² – 88ab + 16b²]
(ii) 36 – x² – y² – 2xy.
(iii) 25x² + 49y² -70xy – 15x + 21y.
(iv) 4a² – 4a – 3.
(v) 64 – x² – y² – 2xy.
(vi) 25a² – (3b + 4c)²
Solution:
(i) The given expression is 100 – [121a² – 88ab + 16b²].
Now, expand the given expression. That is,
100 – [121a² – 88ab + 16b²] = (10)^2 – [(11a)^2 – 2(11a)(4b) + (4b)^2].
By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 – 2xy + y^2.
Here, x = 11a and y = 4b.
So, (x – y)^2= (11a – 4b)^2.
Then, 100 – [121a² – 88ab + 16b²] is equal to (10)^2 – (11a – 4b)^2.
By comparing the above expression, it matches with the expression x^2 – y^2 = (x + y) (x – y).
Here, x = 10 and y = (11a – 4b).
x^2 – y^2 = (x + y) (x – y).
(10)^2 – (11a – 4b)^2 = (10 + (11a – 4b)) (10 – (11a – 4b)).
Then, 100 – [121a^2 – 88ab + 16b^2] is equal to (10 + (11a – 4b)) (10 – (11a – 4b)).
(ii) The given expression is 36 – x² – y² – 2xy.
Now, expand the given expression. That is,
36 – x² – y² – 2xy = (6)^2 – [x^2 + y^2 + 2xy].
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + 2xy + y^2.
Here, x = x and y = y.
So, (x + y)^2= (x + y)^2.
Then, 36 – x² – y² – 2xy is equal to (6)^2 – (x + y)^2.
By comparing the above expression, it matches with the expression x^2 – y^2 = (x + y) (x – y).
Here, x = 6 and y = (x + y).
x^2 – y^2 = (x + y) (x – y).
((6)^2 – (x + y)^2) = (6 + (x + y)) (6 – (x + y)).
Then, 36 – x² – y² – 2xy is equal to (6 + (x + y)) (6 – (x + y)).
(iii) The given expression is 25x^2 + 49y^2 – 70xy – 15x + 21y.
Now, expand the given expression. That is,
25x^2 + 49y^2 – 70xy – 15x + 21y = [(5x)^2 + (7y)^2 – 2(5x)(7y)] – 15x + 21y.
By comparing the given expression with the basic expressions, it matches with the expression (x – y)^2 = x^2 – 2xy + y^2.
Here, x = 5x and y = 7y.
So, (x – y)^2= (5x – 7y)^2.
Then, [(5x)^2 + (7y)^2 – 2(5x)(7y)] – 15x + 21y is equal to (5x – 7y)^2 – 15x + 21y.
(5x – 7y)^2 – 3(5x – 7y).
Factor out the common term from the above expression. That is,
(5x – 7y) (5x – 7y – 3).
So, 25x^2 + 49y^2 – 70xy – 15x + 21y is equal to (5x – 7y) (5x – 7y – 3).
(iv) The given expression is 4a^2 – 4a – 3.
Now, expand the given expression. That is,
4a^2 – 4a – 3 = (2a)^2 – 2(2a) – 3.
Factor out the common terms from the above expression. That is,
(2a)^2 – 2(2a) – 3 = (2a)(2a – 2) – 3.
= (2a – 2) (2a – 3).
4a^2 – 4a – 3 is equal to (2a – 2) (2a – 3).
(v) The given expression is 64 – x^2 – y^2 – 2xy.
Now, expand the given expression. That is,
64 – x^2 – y^2 – 2xy = (8)^2 – [x^2 + y^2 + 2xy].
By comparing the given expression with the basic expressions, it matches with the expression (x + y)^2 = x^2 + 2xy + y^2.
Here, x = x and y = y.
So, (x + y)^2= (x + y)^2.
Then, (8)^2 – [x^2 + y^2 + 2xy] is equal to (8)^2 – (x + y)^2.
By comparing the above expression with the basic expressions, it matches with the expression (x)^2 – (y)^2 = (x + y) (x – y).
Here, x = 8 and y = x + y.
So, (8)^2 – (x + y)^2 = (8 + (x + y)) (8 – (x + y)).
Then, 64 – x^2 – y^2 – 2xy is equal to (8 + (x + y)) (8 – (x + y)).
(vi) The given expression is 25a^2 – (3b + 4c)^2.
Now, expand the expression. That is,
25a^2 – (3b + 4c)^2 = (5a)^2 – (3b + 4c)^2.
By comparing the given expression with the basic expressions, it matches with the expression x^2 – y^2 = (x + y) (x – y).
Here, x = 5a and y = 3b + 4c.
So, x^2 – y^2 = (x + y) (x – y).
(5a)^2 – (3b + 4c)^2 = [5a + (3b + 4c)] [ 5a – (3b + 4c)].
Then, 25a^2 – (3b + 4c)^2 is equal to [5a + (3b + 4c)] [ 5a – (3b + 4c)].