Factorization

The numerical coefficients of given numerical are 2, 12

Greatest common factor of 2, 12 is 2

Common literals appearing in given numerical is x

Smallest power of x in two monomials = 2

Monomials of common literals with smallest power= x²

Hence, the greatest common factor = 2x²

The numerical coefficients of given numerical are 6,18

Greatest common factor of 6, 18 is 6

Common literals appearing in given numerical are x and y

Smallest power of x in both monomials= 2

Smallest power of y in both monomials = 1

Binomials of common literals with smallest power= x²y

Hence, the greatest common factor = 6x²y

The numerical coefficients of given numerical are 7, 21, 14

Greatest common factor of 7, 21, 14 is 7

Common literals appearing in given numerical are x and y

Smallest power of x in three monomials = 1

Smallest power of y in three monomials = 0

Monomials of common literals with smallest power= x

Hence, the greatest common factor = 7x

The numerical coefficients of given numerical are 42 and 63.

Greatest common factor of 42, 63 is 21.

Common literals appearing in given numerical are x, y and z

Smallest power of x in two monomials = 2

Smallest power of y in two monomials = 1

Smallest power of z in two monomials = 1

Monomials of common literals with smallest power= x²yz

Hence, the greatest common factor = 21x²yz

The numerical coefficients of given numerical are 12, 6, 2

Greatest common factor of 12, 6, 2 is 2.

Common literals appearing in given numerical are a and x

Smallest power of x in three monomials = 2

Smallest power of a in three monomials = 1

Monomials of common literals with smallest power= ax²

Hence, the greatest common factor = 2ax²

The numerical coefficients of given numerical are 9, 15, 16, 21

Greatest common factor of 9, 15, 16, 21 is 3.

Common literals appearing in given numerical are x and y

Smallest power of x in four monomials = 1

Smallest power of y in four monomials = 0

Monomials of common literals with smallest power= x

Hence, the greatest common factor = 3x

The numerical coefficients of given numerical are 4, -12, 18.

Greatest common factor of 4, -12, 18 is 2.

Common literals appearing in given numerical are a and b

Smallest power of a in three monomials = 2

Smallest power of b in three monomials = 1

Monomials of common literals with smallest power= a²b

Hence, the greatest common factor = 2a²b

The numerical coefficients of given numerical are 6, 9, 3

Greatest common factor of 6, 9, 3 is 3.

Common literals appearing in given numerical are x and y

Smallest power of x in three monomials = 1

Smallest power of y in three monomials = 2

Monomials of common literals with smallest power= xy²

Hence, the greatest common factor = 3xy²

The numerical coefficients of given numerical are 0

Common literals appearing in given numerical are a and b

Smallest power of a in two monomials = 2

Smallest power of b in two monomials = 2

Monomials of common literals with smallest power= the greatest common factor = a²b²

The numerical coefficients of given numerical are 36, 54, 90

Greatest common factor of 36, 54, 90 is 18.

Common literals appearing in given numerical are a, b and c

Smallest power of a in three monomials = 2

Smallest power of b in three monomials = 0

Smallest power of c in three monomials = 2

Monomials of common literals with smallest power= a²c²

Hence, the greatest common factor = 18a²c²

The numerical coefficients of given numerical are 0

Common literals appearing in given numerical are x and y

Smallest power of x in two monomials = 2

Smallest power of y in two monomials = 0

Monomials of common literals with smallest power= x²

Hence, the greatest common factor = x²

The numerical coefficients of given numerical are 15, -45, -150

Greatest common factor of 15, -45, -150 is 15.

Common literals appearing in given numerical is smallest power of a in three monomials = 1

Monomials of common literals with smallest power= a

Hence, the greatest common factor = 15a

The numerical coefficients of given numerical are 2, 10, 14.

Greatest common factor of 2, 10, 14 is 2.

Common literals appearing in given numerical are x and y

Smallest power of x in three monomials = 1

Smallest power of y in three monomials = 1

Monomials of common literals with smallest power= xy

Hence, the greatest common factor = 2xy

The numerical coefficients of given numerical are 14, 10, 2.

Greatest common factor of 14, 10, 2 is 2.

Common literals appearing in given numerical are x and y

Smallest power of x in three monomials = 2

Smallest power of y in three monomials = 2

Monomials of common literals with smallest power= x²y²

Hence, the greatest common factor = 2x²y²

The highest common factor of three terms = 5a²

=5a²(a² + 2a -3)

The highest common factor of three terms = y

Therefore,

= y(2xz + 3x² +4y)

The highest common factor of three terms = b²

Therefore,

5a²b² + 4b²c² + 12a²b²c² = b²(3a² + 4c² + 12a²c²)

Greatest common factor of the two terms namely 3x and -9 of expression 3x - 9 is 3

3x = 3 × x and -9 = 3 × (-3)

3x - 9 = 3(x - 3)

Greatest common factor of the two terms namely 5x and -15x² of expression 5x - 15x² is 5x -15x²

5x = 5x(1) and -15x²= 5x(-3x)

5x -15x² = 5x(1 - 3x)

Greatest common factor of the two terms namely 20a12b2 and -15a8b4 of expression 20a12b2 - 15a8b4 is 5a8b2

20a12b2 = 5a8b2 (4a4) and - 15a8b⁴ = 5a⁸b² (-3b²)

20a¹²b² - 15a⁸b⁴ = 5a⁸b² (4a⁴ - 3b²)

= 5a⁸b²((2a)² - (b√3)²)

= 5a⁸b²(2a + b√3)(2a - b√3)

Greatest common factor of the two terms namely 72x⁶y⁷ and - 96x⁷y⁶ of expression 72x⁶y⁷ - 96x⁷y⁶ is 24x⁶y⁶

72x⁶y⁷ = 24x⁶y⁶ (3y) and - 96x⁷y⁶ = 24x⁶y⁶(-4x)

72x⁶y⁷ - 96x⁷y⁶ = 24x⁶y⁶ (3y - 4y)

Greatest common factor of the two terms namely 20x³, -40x² and 80x of expression 20x³ - 40x² + 80x is 20x

20x³ - 40x² + 80x= 20x(x² - 2x +4)

Greatest common factor of the two terms namely 2x³y², - 4x²y³, - 8xy⁴ of expression 2x³y² - 4x²y³ - 8xy⁴ is 2xy²

2x³y² - 4x²y³ - 8xy⁴ = 2xy² (x² - 2xy + 4y)

Greatest common factor of the two terms namely 10m³n², 15m⁴n, - 20m²n³ of expression 10m³n² + 15m⁴n - 20m²n³ is 5mn²

10m³n² + 15m⁴n - 20m²n³ = 5mn²(2mn + 3m² - 4n)

Greatest common factor of the two terms namely 2a⁴b⁴, - 3a³b⁵, 4a²b⁵ of expression 2a⁴b⁴ - 3a³b⁵ + 4a²b⁵ is a²b⁴

2a⁴b⁴ - 3a³b⁵ + 4a²b⁵ = a²b⁴ (2a² - 3ab + 4b)

Greatest common factor of the two terms namely 28a², 14a²b², - 21a⁴ of expression 28a² + 14a²b² - 21a⁴ is 7a²

28a² + 14a²b²-21a⁴ = 7a²(4 + 2b² - 3a²)

Greatest common factor of the two terms namely a⁴b, - 3a²b², - 6ab³ of expression a⁴b - 3a²b² - 6ab³ is ab

a⁴b - 3a²b² - 6ab³ = ab (a³ - 3ab -6ab²)

Greatest common factor of the two terms namely 21lmn, - 3lm²n, 4lmn² of expression 21lmn - 3lm²n + 4lmn² is lm

21lmn - 3lm²n + 4lmn² = lm(21 - 3m + 4n)

Greatest common factor of the two terms namely x⁴y², - x²y⁴, - x⁴y⁴ of expression x⁴y² - x²y⁴ - x⁴ y⁴ is x²y²

x⁴y² - x²y⁴ - x⁴y⁴ = x²y² (x² - y² -x²y²)

Greatest common factor of the two terms namely 9x²y and 3axy of expression 9x²y + 3axy is 3xy

9x²y + 3axy = 3xy(3x² +a)

Greatest common factor of the two terms namely 16m - 4m² of expression 16m - 4m² is 4m

16m - 4m² = 4m(4 - m)

Greatest common factor of the two terms namely -4a, 4ab, -4ca of expression -4a + 4ab -4ca is -4a

-4a + 4ab -4ca = -4a(a - b + c)

Greatest common factor of the two terms namely x²yz, xy²z, xyz² of expression x²yz + xy²z + xyz² is xyz

x²yz + xy²z + xyz² = xyz(x + y +z)

Greatest common factor of the two terms namely -4a, 4ab, -4ca of expression -4a + 4ab -4ca is -4a

ax²y + bxy² + cxyz = xy (ax + by + cz)

(6x + 7y) (2x – y) [Therefore, taking (2x – y) common)]

-2r (x – y) + s (x – y) [Therefore, taking – 1 common]

= (x – y) (-2r + s) [Therefore, taking (x – y) common]

= (x – y) (s – 2r)

(7a + 3b) (2x – 3) [Therefore, taking (2x – 3) common]

(9a – 12a²) (6a – 5b) [Therefore, taking (6a – 5b) common]

(x – 2y) [5 (x – 2y) + 3] [Therefore, taking (x – 2y) common]

= (x – 2y) (5x – 10y + 3)

16 (2l – 3m²) + 12 (2l – 3m) [Therefore, 3m – 2l = - (2l – 3m)]

= 4 (2l – 3m) [4 (2l – 3m) + 3] [Therefore, taking 4 (2l – 3m) common]

= 4 (3l – 2m) (8l – 12m + 3)

(3a – b) (x – 2y) [Therefore, taking (x – 2y) as common]

(a² + b² + c²) (x + y) [Therefore, taking (x + y) common in each term]

(x – y) (x – y + 1) [Therefore, taking (x – y) common)

[6 – 4 (a + 2b)] (a + 2b) [Therefore, taking (a + 2b) common]

= (6 – 4a – 8b) (a + 2b)

a (x – y) – 2b (x – y) + c (x – y)² [Therefore, (y – x) = - (x – y)]

= (x – y) [a – 2b + c (x – y)]

= (x – y) (a – 2b + cx – cy)

- (x – 2y) [4 (x – 2y – 8] [Therefore, taking – (x – 2y) as common]

= - (x – 2y) (4x – 8y – 8)

x² (a – 2b) (x + 1) [Therefore, taking x² (a – 2b) as common]

(a + b) (2x – 3y + 3x – 2y) [Therefore, taking (a + b) common]

= (a + b) (5x – 5y)

2 (x + y) [2 (3a – b) + 3 (2b – 3a)] [Therefore, by taking 2 (x + y) common]

= 2 (x + y) (4b – 3a)

q (r + s) – p (r + s)

= (q – p) (r + s)

p (pq – r²) – 1 (pq – r²)

= (p – 1) (pq – r²)

1 (1 + xy) + x (1 + xy)

= (1 + x) (1 + xy)

a (x + y) – b (x + y)

= (a – b) (x + y)

x (a² + b²) – y (a² + b²)

= (x – y) (a² + b²)

x (x + 3) + y (x + 3)

= (x + y) (x + 3)

2a (x + y) + b (x + y)

= (2a + b) (x + y)

a (b – y) – y (b – y)

= (a – y) (b – y)

a (xy – z) + bc (xy – z)

= (a + bc) (xy – z)

2m (m – 1) – n² (m – 1)

= (2m – n²) (m – 1)

y² (1 + x²) + x (1 + x²)

= (x – y²) (1 + x²)

2x (3y – 2) – 3 (3y – 2)

= (2x – 3) (3y – 2)

x (x + b) – 2a (x + b)

= (x – 2a) (x + b)

x (x² + 3y²) – 2y (x² + 3y²)

=(x – 2y) (x² + 3y²)

abx² – ayx – bx – y

= bx (ax – 1) + y (ax – 1)

= (bx + y) (ax – 1)

a²x² + b²y² + 2axby + b²x² + a²y² – 2axby

= a² (x² + y²) + b² (x² + y²)

= (a² + b²) (x² + y²)

8 (a – b)² [2 (a – b) – 3]

= 8 (a – b)² [2a – 2b – 3]

abx² + ab + xa² + xb²

= ax (bx + a) + b (bx + a)

= (ax + b) (bx + a)

a²x² + ax³ + x + a

= x (ax² + 1) + a (ax² + 1)

= (x + a) (ax² + 1)

a² – 2ab – ac + 2bc

= a (a – c) – 2b (a – c)

= (a – 2b) (a – c)

a² + ab + ac – bc

= a (a – c) + b (a – c)

= (a + b) (a – c)

x (x – 1) – 11y (x – 1)

= (x – 11y) (x – 1)

a (b – 1) – 1 (b – 1)

= (a – 1) (b – 1)

x (x – 1) – y (x – 1)

= (x – y) (x – 1)

(4x)² – (5y)²

= (4x + 5y) (4x – 5y)

Consider 27x² - 12y²,

Taking 3 common we get,

3 [(3x)² – (2y)²]

As we know a² - b² = (a-b) (a+b)

= 3 (3x + 2y) (3x – 2y)

(12a)² – (17b)²

= (12a + 17b) (12a – 17b)

3 (4m² – 9)

= 3 [(2m)² – 3²]

= 3 (2m + 3) (2m – 3)

5 (25x² – 9y²)

= 5 [(5x)² – (3y)²]

= 5 (5x + 3y) (5x – 3y)

(12a)² – (13b)²

= (12a + 13b) (12a – 13b)

(2a – b)² – (4c)²

= (2a – b + 4c) (2a – b – 4c)

(x + 2y)² – [2 (2x – y)]²

= [(x + 2y) + 2 (2x – y)] [x + 2y – 2 (2x – y)]

= (x + 4x + 2y – 2y) (x – 4x + 2y + 2y)

= (5x) (4y – 3x)

3a³ (a² – 16)

= 3a³ (a² – 4²)

= 3a³ (a + 4) (a – 5)

(a²)² – (4b²)²

= (a² + 4b²) (a² – 4b²)

(x⁴)²–(1)²

= (x⁴ + 1) (x⁴ – 1)

8² – (a + 1)²

= [8 + (a + 1)] [8 – (a + 1)]

= (a + 9) (7 – a)

(6l)² – (m + n)²

= (6l + m + n) (6l – m – n)

(5x²y²)² – (1)²

= (5x²y² – 1) (5x²y² + 1)

(a²)² – ()²

= (a² + ) (a² - )

x [x² – (12)²]

= x (x + 12) (x – 12)

(x – 4y)² – (25)²

= (x – 4y + 25) (x – 4y – 25)

[3 (a – b)]² – [10 (x – y)]²

= [3 (a – b) + 10 (x + y)] [3 (a – b) – 10 (x – y)]

= [3a – 3b + 10x – 10y] [3a – 3b – 10x + 10y]

(3 + 2a)² – (5a)²

= (3 + 2a + 5a) (3 + 2a – 5a)

= (7a + 3) (3 – 3a)

[(x + y) + (a – b)] [(x + y) – (a – b)]

= (x + y + a – b) (x + y – a + b)

(xy)² – (yz)²

= ( + ) ( - )

= y² ( + z) ( - z)

3ab² (25a² – 36b²)

= 3ab² [(5a)² – (6b)²]

= 3ab² (5a + 6b) (5a – 6b)

x³ (x² – 16)

= x³ (x² – 4²)

= x³ (x + 4) (x – 4)

2 (- )

= 2 [()² – ()²]

= 2 (+ ) ( - )

x (256x⁴ – 81)

= x [(16x²)² – 9²]

= x (16x + 9) (16x – 9)

(a²)² – [(2b + c)²]²

= [a² + (2b + c)²] [a² – (2b + c)²]

= [a² + (2b + c)²] [a + 2b + c] [a – 2b – c]

[(3x + 4y)²]² – (x²)²

= [(3x + 4y)² + x²] [(3x + 4y)² – x²]

= [(3x + 4y)² + x²] [3x + 4y + x] [3x + 4y – x]

(pq)² – (p²q²)²

= (pq + p²q²) (pq – p²q²)

= (pq)² (1 + pq) (1 – pq)

3xy (x² – 81y²)

= 3xy [x² – (9y)²]

= (3xy) (x + 9y) (x – 9y)

(a²b²)² – (4c²)²

= (a²b² + 4c²) (a²b² – 4c²)

= (a²b² + 4c²) (ab + 2c) (ab – 2c)

(x²)² – (25)²

= (x² + 25) (x² – 25)

= (x² + 25) (x + 5) (x – 5)

(x²)² – (1)²

= (x² + 1) (x² – 1)

= (x² + 1) (x + 1) (x – 1)

[7 (a – b)]² – [5 (a + b)]²

= [7 (a – b) + 5 (a + b)] [7 (a – b) – 5 (a + b)]

= (7a – 7b + 5a + 5b) (7a – 7b – 5a – 5b)

= (12a – 2b) (2a – 12b)

= 2 (6a – b) 2 (a – 6b)

= 4 (6a – b) (a – 6b)

x – y – (x² – y²)

= x – y – (x + y) (x – y)

= (x – y) (1 – x – y)

[4 (2x – 1)]² – (5y)²

= (8x – 4 + 5y) (8x – 4 – 5y)

[2x (xy + 1)]² – [3 (x – 1)]²

= (2xy + 2 + 3x – 3) (2xy + 2 – 3x + 3)

= (2xy + 3x – 1) (2xy – 3x + 5)

(2x + 1)² – (3x²)²

= (2x + 1 + 3x²) (2x + 1 – 3x²)

= (3x² + 2x + 1) (-3x² + 2x + 1)

(x²)² – (2y – 3z)²

= (x² + 2y – 3z) (x² – 2y + 3z)

(a + b) (a – b) + (a – b)

= (a – b) (a + b + 1)

(4a²)² – (b²)²

= (4a² + b²) (4a² – b²)

= (4a² + b²) (2a + b) (2a – b)

(a²)² – [4 (b – c)²]

= [a² + 4 (b – c)²] [a² – 4 (b – c)²]

= [a² + 4 (b – c)²] [(a + 2b – 2c) (a – 2b + 2c)]

2a (a⁴ – 16)

= 2a [(a)² – (4)²]

= 2a (a² + 4) (a² – 4)

= 2a (a² + 4) (a + 2) (a – 2)

(a²b²)² – (9c²)²

= (a²b² + 9c²) (a²b² – 9c²)

= (a²b² + 9c²) (ab + 3c) (ab – 3c)

xy (y⁸ – x⁸)

= xy [(y⁴)² – (x⁴)²]

= xy (y⁴ + x⁴) (y⁴ – x⁴)

= xy (y⁴ + x⁴) (y² + x²) (y² – x²)

= xy (y⁴ + x⁴) (y² + x²) (y + x) (y – x)

x (x² – 1)

= x (x + 1) (x – 1)

2 [(3ax)² – (4)²]

= 2 (3ax + 4) (3ax – 4)

4x² + 12xy + 9y²

= (2x)² + (3y)² + 2 (2x) (3y)

= (2x + 3y)²

Consider 9a² – 24ab + 16b²,

As we know (x - y)² = x² + y² - 2xy

Here x = 3a, y = 4b

So,

(3a)² + (4b)² – 2 (3a) (4a)

= (3a – 4b)²

(pq)² + (3r)² – 2 (pq) (3r)

= (pq – 3r)²

9 (4a² + 4a + 1)

= 9 [(2a)² + 2 (2a) + 1¹]

= 9 (2a + 1)²

(a + b)² - 4²

= (a + b + 4) (a + b – 4)

(3z)² – [x² – 2 (x) (2y) + (2y)²]

= (3z)² – (x – 2y)²

= [3z + (x – 2y)] [3z – (x – 2y)]

(3a²)² – 2 (4a²) (3b²) + (4b²)² – (16)²

= (3a² – 4b²)² – (16)²

= (3a² – 4b² + 16) (3a² – 4b² – 16)

4² – [(a³)² – 2 (a³) (2b³) + (2b³)²]

= 4² – (a³ – 2b³)²

= [4 + (a³ – 2b³)] [4 – (a³ – 2b³)]

(a + b)² – c²

= (a + b + c) (a + b – c)

(x + 1)² – (3y)²

= (x + 3y + 1) (x – 3y + 1)

a² + ab + 3ab + 3b²

= a (a + b) + 3b (a + b)

= (a + 3b) (a + b)

-x² – 4x + 96

= -x² – 12x + 8x + 96

= -x (x + 12) + 8 (x + 12)

= (x + 12) (-x + 8)

(a²)² + (a²)² + 2 (2a²) + 4 – a²

= (a² + 2)² + (-a²)

= (a² + 2 + a) (a² + 2 – a)

(2x²)² + 1 + 4x² – 4x²

= (2x² + 1)² – 4x²

= (2x² + 2x + 1) (2x² – 2x + 1)

(2x²)² + (y²)² + 4x²y² – 4x²y²

= (2x² + y²)² – 4x²y²

= (2x² + y² + 2xy) (2x² + y² – 2xy)

x² + 4 + 4x – 6x – 12 + 9

= x² + 1 – 2x

= (x – 1)²

25 – (p² + q² + 2pq)

= (5)² – (p + q)²

= (5 + p + q) (5 –p – q)

= - (p + q – 5) (p + q + 5)

(x – 3y)² – (5a)²

= (x – 3y + 5a) (x – 3y – 5a)

49 – (a² – 8ab + 16b²)

= 49 – (a – 4b)²

= (7 + a – 4b) (7 – a + 4b)

= - (a – 4b + 7) (a – 4b – 7)

(a – 4b)²- (5c)²

= (a – 4b + 5c) (a – 4b – 5c)

x² + 6y – (y² – 6y + 9)

= x² – (y – 3)²

= (x + y – 3) (x – y + 3)

(5x)² – 2 (5x) + 1 – (6y)²

= (5x – 1)² – (6y)²

= (5x – 1 + 6y) (5x – 1 – 6y)

a² – (b² – 2bc + c²)

= a² – (b – c)²

= (a + b – c) (a – b + c)

(a + b)² – c²

= (a + b + c) (a + b – c)

49 – (x² + y² – 2xy)

= 7² – (x – y)²

= [7 + (x – y)] [7 – x + y]

a² – 2 (a) (2b) + (2b)² – (2c)²

= (a – 2b)² – (2c)²

= (a – 2b + 2c) (a – 2b – 2c)

x² – 2 (x) (2z) + (2z)² – y²

As (a-b)² = a² + b² – 2ab

= (x – 2z)² – y²

= (x – 2z + y) (x – 2z – y)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = 12, pq = -45

Clearly,

15 – 3 = 12, 15 (-3) = -45

Therefore, split 12x as 15x – 3x

Therefore,

x² + 12x – 45 = x² + 15x – 3x – 45

= x (x + 15) – 3 (x + 15)

= (x – 3) (x + 15)

- (x² – 3x – 40)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = - 3, pq = - 40

Clearly,

5 – 8 = -3, 5 (-8) = -40

Therefore, split -3x as 5x – 8x

Therefore,

x² - 3x – 40 = x² + 5x – 8x – 40

= x (x + 5) – 8 (x + 5)

= (x – 8) (x + 5)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = 3, pq = -88

Therefore, split 3a as 11a – 8a

Therefore,

a² + 3a – 88 = a² + 11a – 8a – 88

= a (a + 11) – 8 (a + 11)

= (x – 8) (a + 11)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = -14, pq = -51

Clearly,

3 – 17 = -14, 3 (-17) = -51

Therefore, split 14a as 3a – 17a

Therefore,

a² – 14a – 51 = a² + 3a – 17a – 51

= a (a + 3) – 17 (a + 3)

= (a – 17) (a + 3)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = 14, pq = 45

Clearly,

5 + 9 = 14, 5 (9) = 45

Therefore, split 14x as 5x + 9x

Therefore,

x² + 14x + 45 = x² + 5x + 9x + 45

= x (x + 5) – 9 (x + 5)

= (x + 9) (x + 5)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = -22, pq = 120

Clearly,

-12 – 10 = -22, (-12) (-10) = -120

Therefore, split -22x as -12x – 10x

Therefore,

x² - 22x + 120 = x² - 12x – 10x + 120

= x (x - 12) – 10 (x - 12)

= (x – 10) (x - 12)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = -11, pq = -42

Clearly,

3 – 14 = -11, 3 (-14) = -42

Therefore, split (-11x) as 3x – 14x

Therefore,

x² - 11x – 42 = x² + 3x – 14x – 42

= x (x + 3) – 14 (x + 3)

= (x – 14) (x + 3)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = 2, pq = -3

Clearly,

p = 3, q = -1

Therefore, split (2a) as (3a – a)

Therefore,

a² + 2a – 3 = a² + 3a – a – 3

= a (a + 3) – 1 (a + 3)

= (a – 1) (a + 3)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = 14, pq = 48

Clearly,

8 + 6 = 14, 8 (6) = 48

Therefore, split (14a) as 8a + 6a

Therefore,

a² + 14a + 48 = a² + 8a + 6a + 48

= a (a + 8) + 6 (a + 8)

= (a + 6) (a + 8)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = -4, pq = -21

Clearly,

3 – 7 = -4, 3 (-7) = -21

Therefore, split (-4x) as 3x – 7x

Therefore,

x² + 4x – 21 = x² + 3x – 7x – 21

= x (x + 3) – 7 (x + 3)

= (x – 7) (x + 3)

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = 5, pq = -36

Clearly,

9 – 4 = 5, 9 (-4) = -36

Therefore, split 5y as 9y – 4y

Therefore,

y² + 5y – 36 = y² + 9y – 4y – 36

= y (y + 9) – 4 (y + 9)

= (y – 4) (y + 9)

It can be written as (a² – 5a)² - 6²

Using a² – b² = (a + b) (a – b)

(a² – 5a)² – 6² = (a² – 5a + 6) (a² – 5a – 6)

To factorize (a² – 5a + 6), we need to find p and q where,

p + q = -5, pq = 6

Clearly,

-2 – 3 = -5, (-2) (-3) = 6

Therefore, split -5a as a – 6a

Therefore,

a² -5a – 6 = a² - a – 6a + 6

= (a – 6) (a – 1)

Therefore,

(a² – 5a)² – 3b = (a² – 5a + b) (a² – 5a – 6)

= (a – 1) (a – 2) (a – 3) (a – 6)

a² – 3a – 54

In order to factorize the given expression, we find to find two numbers p and q such that:

p + q = -3, pq = -54

Clearly,

6 – 9 = - 3, 6 (-9) = -54

Therefore, split – 3a as 6a – 9a

Therefore,

a² – 3a – 54 = a² + 6a – 9a – 54

= (a - 9) (a + 6)

Therefore,

(a + 7) (a – 10) + 16 = (a – 9) (a + 6)

Here, coefficient of x² = 2, coefficient of x = 5and constant term = 3

We shall now split up the coefficient of x i.e., 5 into two parts whose sum is 5 and product is 2 * 3 = 6

So, we write middle term 5x as 2x + 3x

Thus, we have

2x² + 5x + 3 = 2x² + 2x + 3x + 3

= 2x (x + 1) + 3 (x + 1)

= (2x + 3) (x + 1)

Here, coefficient of x² = 2, coefficient of x = - 3 and constant term = -2

We shall now split up the coefficient of x i.e., -3 into two parts whose sum is -3 and product is 2 * -2 = - 4

So, we write middle term -3x as -4x + x

Thus, we have

2x² - 3x – 2 = 2x² - 4x + x – 2

= 2x (x – 2) + 1 (x – 2)

= (x – 2) (2x + 1)

Here, coefficient of x² = 3, coefficient of x = 10 and constant term = 3

We shall now split up the coefficient of x i.e., 10 into two parts whose sum is 10 and product is 3 * 3 = 9

So, we write middle term 10x as 9x + x

Thus, we have

3x² + 10x + 3 = 3x² + 9x + x + 3

= 3x (x + 3) + 1 (x + 3)

= (3x + 1) (x + 3)

= - 2x² + 7x – 6

Here, coefficient of x² = -2, coefficient of x = 7and constant term = -6

We shall now split up the coefficient of x i.e., 7 into two parts whose sum is 7 and product is -2 * -6 = 12

Clearly,

4 + 3 = 7 and,

4 * 3 = 12

So, we write middle term 7x as 4x + 3x

Thus, we have

-2x² + 7x – 6 = -2x² + 4x + 3x – 6

= -2x (x – 2) + 3 (x – 2)

= (x – 2) (3 – 2x)

Here, coefficient of x² = 7, coefficient of x = -19 and constant term = -6

We shall now split up the coefficient of x i.e., -19 into two parts whose sum is -19 and product is 7 * -6 = -42

Clearly,

2 - 21 = -19 and,

2 * (-21) = - 42

So, we write middle term - 19x as 2x - 21x

Thus, we have

7x² - 19x – 6 = 7x² + 2x - 21x – 6

= x (7x + 2) - 3 (7x + 2)

= (7x + 2) (x – 3)

28 – 31x – 5x² = - 5x² – 31x + 28

Here, coefficient of x² = -5, coefficient of x = - 31 and constant term = 28

We shall now split up the coefficient of x i.e., - 31 into two parts whose sum is - 31 and product is -5 (28) = - 140

Clearly,

4 - 35 = - 31 and,

4 (-35) = - 140

So, we write middle term - 31x as 4x - 35x

Thus, we have

– 5x² – 31x + 28 = -5x² + 4x - 35x + 28

= -x (5x – 4) - 7 (5x – 4)

= - (x + 7) (5x - 4)

3 + 23y – 8y² = - 8y² + 23y + 3

Here, coefficient of y² = -8, coefficient of y = 23 and constant term = 3

We shall now split up the coefficient of x i.e., 23 into two parts whose sum is 23 and product is -8 (3) = - 24

Clearly,

24 - 1 = 23 and,

24 (-1) = - 24

So, we write middle term 23y as 24y - y

Thus, we have

-8y² + 23y + 3 = - 8² + 24y - y + 3

= -8y (y – 3) - 1 (y – 3)

= - (8y + 1) (y – 3)

11x² – 54x + 63

Here, coefficient of x² = 11, coefficient of x = - 54 and constant term = 63

We shall now split up the coefficient of x i.e., -54 into two parts whose sum is - 54 and product is 11 * 63 = 693

Clearly,

-33x - 21x = - 54x and,

(-33) * (-21) = 693

So, we write middle term - 54x as - 33x - 21x

Thus, we have

11x² – 54x + 63 = 11x² - 33x - 21x – 6

= 11x (x – 3) - 21 (x – 3)

= (11x – 21) (x – 3)

7x – 6x² + 20 = - 6x² + 7x + 20

Here, coefficient of x² = -6, coefficient of x = 7and constant term = 20

We shall now split up the coefficient of x i.e., 7 into two parts whose sum is 7 and product is -6 * 20 = - 120

Clearly,

15 - 8 = 7 and,

15 (-8) = - 120

So, we write middle term 7x as 15x - 8x

Thus, we have

-6x² + 7x + 20 = -6x² + 15x - 8x + 20

= -3x (2x – 5) - 4 (2x – 5)

= - (3x + 4) (2x - 5)

3x² + 22x + 35

Here, coefficient of x² = 3, coefficient of x = 22 and constant term = 35

We shall now split up the coefficient of x i.e., 22 into two parts whose sum is 22 and product is 3 * 35 = 105

So, we write middle term 22x as 15x + 7x

Thus, we have

3x² + 22x + 35= 3x² + 15x + 7x + 35

= 3x (x + 5) + 7 (x + 5)

= (3x + 7) (x+ 5)

12x² – 17xy + 6y²

Here, coefficient of x² = 12, coefficient of x = -17and constant term = 6y²

We shall now split up the coefficient of middle term i.e., -17y into two parts whose sum is -17y and product is 12 * 6y² = 72y²

Clearly,

-9y – 8y = -17y and,

(-9y) (-8y) = 72y²

So, we replace middle term -17xy = - 9xy – 8xy

Thus, we have

12x² -17xy+ 6y² = 12x² - 9xy - 8xy + 6y²

= 3x (4x – 3y) – 2y (4x – 3y)

= (3x – 2y) (4x – 3y)

Here, coefficient of x² = 6, coefficient of x = -5y and constant term = - 6y²

We shall now split up the coefficient of middle term i.e., -5y into two parts whose sum is -5y and product is 6 (-6y²) = - 36y²

Clearly,

4y – 9y = -5y and,

(4y) (-9y) = - 36y²

So, we replace middle term -5xy = 4xy – 9xy

Thus, we have

6x² -5xy- 6y² = 6x² + 4xy - 9xy - 6y²

= (2x – 3y) (3x + 2y)

Here, coefficient of x² = 6, coefficient of x = -13y and constant term = 2y²

We shall now split up the coefficient of middle term i.e., -13y into two parts whose sum is -13y and product is 6 (2y²) = 12y²

Clearly,

-12y – y = -13y and,

(-12y) (-y) = 12y²

So, we replace middle term -13xy = -12xy – xy

Thus, we have

6x² -13xy+ 2y² = 6x² - 12xy - xy - 2y²

= (6x – y) (x - 2y)

Here, coefficient of x² = 14, coefficient of x = 11y and constant term = - 15y²

We shall now split up the coefficient of middle term i.e., 11y into two parts whose sum is 11y and product is 14 (-15y²) = - 210y²

Clearly,

21y – 10y = 11y and,

(21y) (-10y) = - 210y²

So, we replace middle term 11xy = 21xy – 10xy

Thus, we have

14x² + 11xy- 15y² = 14x² + 21xy - 10xy - 15y²

= 2x (7x – 5y) + 3y (7x – 5y)

= (2x + 3y) (7x - 5y)

Here, coefficient of a² = 6, coefficient of a = 17b and constant term = - 3b²

We shall now split up the coefficient of middle term i.e., 17b into two parts whose sum is 17b and product is 6 (-3b²) = - 18b²

Clearly,

18b – b = 17b and,

6 (-3b²) = - 36y²

So, we replace middle term 17ab = 18ab – ab

Thus, we have

6a² +17ab– 3b² = 6a² + 18ab - ab – 3b²

= 6a (a + 3b) – b (a + 3b)

= (6a – b) (a + 3b)

Here, coefficient of a² = 36, coefficient of a = 12bc and constant term = - 15b²c²

We shall now split up the coefficient of middle term i.e., 12bc into two parts whose sum is 12bc and product is 36 (-15b²c²) = - 500b²c²

So, we replace middle term 12abc = 30abc – 18abc

Thus, we have

36a² –12abc– 15b²c² = 36a² + 30abc – 18abc – 15b²c²

= (6a + 5bc) (6a – 3bc)

Here, coefficient of x² = 15, coefficient of x = -16yz and constant term = - 15y²z²

We shall now split up the coefficient of middle term i.e., -16yz into two parts whose sum is -16yz and product is 15 (-15y²z²) = - 225y²z²

Clearly,

-25yz + 9yz = -16yz and,

(-25yz) (9yz) = - 225y²z²

So, we replace middle term -16xyz = -25yz – 9yz

Thus, we have

15x² -16xyz- 15y²z² = 15x² - 25yz + 9yz - 15y²z²

= 5x (3x – 5yz) + 3yz (3x – 5yz)

= (5x + 3yz) (3x - 5yz)

x² + 4y² – 4xy – 5x + 10y + 6

Here, coefficient of (x – 2y)² = 1, coefficient of (x – 2y ) = -5 and constant = 6

We shall now split up the coefficient of middle term i.e., -5 into two parts whose sum is -5 and product is 6 (1) = 6

Clearly,

-2 - 3 = -5 and,

-2 (-3) = 6

So, we replace-5 (x – 3y) = -2 (x – 2y) – 3 (x – 2y)

Thus, we have

(x – 2y)² – 5 (x – 2y) + 6 = (x – 2y)² – 2 (x – 2y) – 3 (x – 2y) + 6

= (x – 2y - 2) (x - 2y - 3)

Here, coefficient of (2a – b)² = 1, coefficient of (2a – b) = 2 and constant term = - 8

We shall now split up the coefficient of middle term i.e., 2 into two parts whose sum is 2 and product is -8 (1) = - 8

Clearly,

4 - 2 = 2 and,

4 (-2) = - 8

So, we replace 2 (2a – b) = 4 (2a –b) – 2 (2a – b)

Thus, we have

(2a – b)² + 2 (2a – b) – 8 = (2a – b)² + 4 (2a – b) – 2 (2a – b) - 8

= (2a – b) (2a – b + 4) – 2 (2a – b + 4)

= (2a – b – 2) (2a – b + 4)

p² + 6p + 8

Here, coefficient of p² is unity so we add and subtract square of half of coefficient of p

Therefore,

p² + 6p + 8 = p² + 6p + 3² – 3² + 8 (Adding and subtracting 3²)

= (p + 3)² – 1² (By completing the square)

= (p + 3 – 1) (p + 3 + 1)

= (p + 2) (p + 4)

q² – 10q + 21 Coefficient of q² is 1 so we add and subtract square of half of coefficient of q

Therefore,

q² – 10q + 21 = q² – 10q+ 5² – 5² + 21 (Adding and subtracting 5²)

= (q – 5)² – 2² (By completing the square)

= (q – 5 – 2) (q – 5 + 2)

= (q – 7) (q – 3)

4y² + 12y + 5

We have 4y² + 12y + 5 = 4 (y² + 3y + ) [Therefore, coefficient of y² = 1]

= 4 [y² + 3y + ()² – ()² + ]

= 4 [(y + )² – 1²] (Completing the square)

= 4 (y + + 1) (y + – 1)

= (2y + 5) (2y + 1)

p² + 6p – 16

Coefficient of p² = 1

Therefore, we have

p² + 6p + 3² – 3² – 16 (Adding and subtracting 3²)

= (p + 3)² – 5² (Completing the square)

= (p + 3 + 5) (p + 3 – 5)

= (p + 8) (p – 2)

x² + 12x + 20

Coefficient of x² = 1

Therefore, we have

x² + 12x + 6² – 6² + 20 (Adding and subtracting 6²)

= (x + 6)² – 4² (Completing the square)

= (x + 6 + 4) (x + 6 – 4)

= (x + 10) (x + 2)

= 4 [x - + 1] [x - – 1]

= (2x – 1) (2x – 5)

a² – 14a – 51

Coefficient of a² = 1

Therefore, we have

a² – 14a – 51 = a² – 14a + 7² – 7² – 51 (Therefore, adding and subtracting 7²)

= (a – 7)² – 10² (Completing the square)

= (a – 7 + 10) (9 – 7 – 10)

= (a + 3) (a – 17)

a² + 2a – 3

Coefficient of a² = 1

Therefore, we have

a² + 2a – 3 = a² + 2a + 1² – 1² – 3 (Adding and subtracting 1²)

= (a + 1)² – 2² (Completing the square)

= (a + 1 + 2) (a + 1 – 2)

= (a + 3) (a – 1)

4x² – 12x + 5

We have,

4x² – 12x + 5 = 4 (x² – 3x + )

= 4 [x² – 3x + ()² – ()² + )] [Therefore, adding and subtracting ()²]

= 4 [(x - )² – 1²] (Therefore, completing the square)

y² – 7y + 12

Coefficient of y² = 1

Therefore, we have

y² – 7y + 12 = y² – 7y + ()² – ()² + 12 [By adding and subtracting ()²]

= (y - )² – ()² (Completing the square)

= (y - - ) (y - + )

= (y – 4) (y – 3)

z² – 4z – 12

Coefficient of z² = 1

Therefore, we have

z² – 4z – 12 = z² – 4z + 2² – 2² – 12 [By adding and subtracting 2²]

= (z – 2)² – 4² (Completing the square)

= (z – 2 + 4) (z – 2 – 4)

= (z + 2) (z – 6)