FACTORING TRINOMIALS WHEN A IS NOT 1

Factoring trinomials when a is not 1.

Problem 1:    

7m² + 6m - 1

Solution :

= 7m² + 6m - 1

= 7m² + 7m – m – 1

= 7m (m + 1) – 1 (m + 1)

= (7m - 1) (m + 1)

Problem 2 :    

3k² - 10k + 7

Solution :

= 3k² - 10k + 7

= 3k² - 3k - 7k + 7

= 3k(k - 1) - 7(k - 1)

= (3k - 7) (k - 1)

Problem 3 :    

5x² - 36x - 81

Solution :

= 5x² - 36x - 81

= 5x² - 45x + 9x – 81

= 5x(x - 9) + 9(x - 9)

= (5x + 9) (x - 9)

Problem 4 :    

2x² - 9x - 81

Solution :

= 2x² - 9x - 81

= 2x² - 18x + 9x - 81

= 2x(x - 9) + 9(x - 9)

= (2x + 9) (x - 9)

Problem 5 :    

3n² - 16n + 20

Solution :

= 3n² - 16n + 20

= 3n² - 6n – 10n + 20

= 3n(n - 2) – 10(n - 2)

= (3n - 10) (n - 2)

Problem 6 :    

2r² + 7r - 30

Solution :

= 2r² + 7r - 30

= 2r² +12r – 5r – 30

= 2r(r + 6) – 5(r + 6)

= (2r - 5) (r + 6)

Problem 7 :    

5k² + 8k + 80

Solution :

5k² + 8k + 80 is not factorable.

Problem 8 :

5x² - 14x + 8

Solution :

= 5x² - 14x + 8

= 5x² - 10x – 4x + 8

= 5x(x - 2) – 4(x - 2)

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

Problem 9 :

7p² - 20p + 12

Solution :

= 7p² - 20p + 12

= 7p² - 14p – 6p + 12

= 7p(p - 2) – 6(p - 2)

= (7p - 6) (p - 2)

Problem 10 :  

3v² + 14v - 49

Solution :

= 3v² + 14v - 49

= 3v² + 21v – 7v – 49

= 3v(v + 7) – 7(v + 7)

= (3v - 7) (v + 7)

Problem 11 :  

7x² - 26x - 45

Solution :

= 7x² - 26x - 45

= 7x² - 35x + 9x – 45

= 7x(x - 5) + 9(x - 5)

= (7x + 9) (x - 5)

Problem 12 :

5p² - 52p + 20

Solution :

= 5p² - 52p + 20

= 5p² - 50p – 2p + 20

= 5p(p - 10) – 2(p - 10)

= (5p - 2) (p - 10)

Problem 13 :

The volume (in cubic feet) of a room in the shape of a rectangular prism is represented by 12z³ − 27z. Find expressions that could represent the dimensions of the room.

Solution :

= 12z³ − 27z

= 3z (4z² - 9)

= 3z (2²z² - 3²)

= 3z [(2z)² - 3²]

Using algebraic identity, we get

= 3z(2z + 3)(2z - 3)

• The length = 3z
• width = 2z + 3
• height = 2z - 3

Problem 14 :

Factor

(a) 3x³ + 6x² − 18x

(b) 7x⁴ − 28x²

Solution :

(a) 3x³ + 6x² − 18x

= 3x(x² + 2x - 3)

= 3x(x² + 3x - 1x - 3)

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

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

So, the factors are 3x (x + 3) and (x - 1).

(b) 7x⁴ − 28x²

= 7x² (x² - 4)

= 7x² (x² - 2²)

= 7x² (x + 2)(x - 2)

So, the factors are 7x² (x + 2) and (x - 2)

Problem 15 :

Can you use the perfect square trinomial pattern to factor y² + 16y + 64? Explain.

Solution :

= y² + 16y + 64

= y² + 8y + 8y + 64

= y(y + 8) + 8(y + 8)

= (y + 8)(y + 8)

= (y + 8)²

Problem 16 :

Which polynomial does not belong with the other three? Explain your reasoning.

a)  g² − 6g + 9      b)  n² − 4     c)  r² + 12r + 36     d)  g² + 25

Solution :