GCF WITH NEGATIVES AND SPECIAL CASE POLYNOMIALS

Factor each completely.

Problem 1 :

16n² - 9

Solution :

16n² - 9

= (4²n² - 3²)

= ((4n)² - 3²)

(a² - b²) = (a + b) (a - b)

= (4n + 3) (4n - 3)

Problem 2 :

4m² - 25

Solution :

= 4m² - 25

= (2²m² - 5²)

= ((2m)² - 5²)

(a² - b²) = (a + b) (a - b)

= (2m + 5) (2m - 5)

Problem 3 :

16b² - 40b + 25

Solution :

= 16b² - 40b + 25

= 16b² - 20b - 20b + 25

= 4b(4b - 5) - 5(4b - 5)

=  (4b - 5) (4b - 5)

= (4b - 5)²

Problem 4 :

4x² - 4x + 1

Solution :

= 4x² - 4x + 1

= 4x² - 2x - 2x + 1

= 2x(2x - 1) - 1(2x - 1)

= (2x - 1) (2x - 1)

= (2x - 1)²

Problem 5 :

9x² - 1

Solution :

= 9x² - 1

= (3²x² - 1²)

= ((3x)² - 1²)

(a² - b²) = (a + b) (a - b)

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

Problem 6 :

n² - 25

Solution :

= n² - 25

= (n² - 5²)

(a² - b²) = (a + b) (a - b)

= (n + 5) (n - 5)

Problem 7 :

n⁴ - 100

Solution :

= n⁴ - 100

= ((n²)² - 10²)

(a² - b²) = (a + b) (a - b)

(n² + 10) (n² - 10)

Problem 8:

a⁴ - 9

Solution :

= a⁴ - 9

= ((a²)² - 3²)

(a² - b²) = (a + b) (a - b)

= (a² + 3) (a² - 3)

Problem 9 :

k⁴ - 36

Solution :

= k⁴ - 36

= ((k²)² - 6²)

(a² - b²) = (a + b) (a - b)

= (k² + 6) (k² - 6)

Problem 10 :

n⁴ - 49

Solution :

= n⁴ - 49

= ((n²)² - 7²)

(a² - b²) = (a + b) (a - b)

(n² + 7) (n² - 7)

Problem 11 :

98n² - 200

Solution :

= 98n² - 200

= 2(49n² - 100)

= 2(7²n²-10²)

= 2((7n)² - 10²)

(a² - b²) = (a + b) (a - b)

= 2(7n + 10) (7n - 10)

Problem 12 :

3 + 6b + 3b²

Solution :

= 3b² + 6b + 3

= 3(b² + 2b + 1)

= 3(b + 1)²

Problem 13 :

400 - 36v²

Solution :

= 400 - 36v²

= 4(100 - 9v²)

= 4(10² - 3²v²)

= 4(10² - (3v)²)

(a² - b²) = (a + b) (a - b)

4(10 + 3v) (10 - 3v)

Problem 14 :

100x² + 180x + 81

Solution :

100x² + 180x + 81

100x² + 90x + 90x + 81

10x(10x + 9) + 9(10x + 9)

(10x + 9) (10x + 9)

(10x + 9)²

Problem 15 :

10n² + 100n + 250

Solution :

10n² + 100n + 250

10(n2 + 10n + 25)

10(n² + 5n + 5n + 25)

10n(n + 5) + 5(n + 5)

10(n + 5) (n + 5)

10(n + 5)²

Problem 16 :

49n² - 56n + 16

Solution :

49n² - 56n + 16

49n² - 28n - 28n + 16

7n(7n - 4) - 4(7n - 4)

(7n - 4) (7n - 4)

(7n - 4)²

Problem 17 :

49x² - 100

Solution :

49x² - 100

(7²x² - 10²)

((7x)² - 10²)

(a² - b²) = (a + b) (a - b)

(7x + 10) (7x - 10)

Problem 18 :

1 - r²

Solution :

1 - r² 

1² - r²

(a² - b²) = (a + b) (a - b)

(1 + r) (1 - r)

Problem 19 :

10p³ - 1960p

Solution :

10p³ - 1960p

10p(p² - 196)

10p(p² - 14²)

(a² - b²) = (a + b) (a - b)

10p(p + 14) (p - 14)

Problem 20 :

343b² - 7b⁴

Solution :

343b² - 7b⁴

7b²(49 - b²)

7b²(7² - b²)

(a² - b²) = (a + b) (a - b)

7b²(7 + b) (7 - b)

Problem 21 :

81v⁴ - 900v²

Solution :

81v⁴ - 900v²

9v²(9v² - 100)

9v²(3²v²-10²)

9v²((3v)² - 10²)

(a² - b²) = (a + b) (a - b)

9v²(3v + 10) (3v - 10)

Problem 22 :

200m⁴ + 80m³ + 8m²

Solution :

200m⁴ + 80m³ + 8m²

8m²(25m² + 10m + 1)

8m²(25m² + 5m + 5m + 1)

8m² 5m(5m + 1) + 1(5m + 1)

8m² (5m + 1) (5m + 1)

8m² (5m + 1)²