FACTOR THE POLYNOMIAL INTO LINEAR FACTORS

Problem 1 :

a³ + 3a² - a – 3

Solution :

= a³ + 3a² - a – 3

Grouping the first two terms and last two terms.

= (a³ + 3a²) + (- a – 3)

= a² (a + 3) – 1(a + 3)

= (a² - 1) (a + 3)

= (a² - 1²) (a + 3)

Using the algebraic identity, we get

= (a + 1) (a - 1) (a + 3)

Problem 2 :

5x² - 15x + 10

Solution :

= 5x² - 15x + 10

= 5x² - 10x – 5x + 10

= 5x(x - 2) – 5(x - 2)

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

Factoring 5, we get

= 5(x - 1) (x - 2)

Problem 3 :

b³ - 4b

Solution :

= b³ - 4b

= b (b² - 4)

(a² - b²) = (a + b) (a - b)

= b(b + 2) (b - 2)

Problem 4 :

4ax² + 4ax – 24a

Solution :

= 4ax² + 4ax – 24a

= 4a(x² + x - 6)

= 4a(x² - 2x + 3x - 6)

= 4a [x(x - 2) + 3(x - 2)]

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

Problem 5 :

12c² - 3

Solution :

= 12c² - 3

= 3(4c² - 1)

(a² - b²) = (a + b) (a - b)

= 3(2c + 1) (2c - 1)

Problem 6 :

x⁴ – 81

Solution :

= x⁴ – 81

(a² - b²) = (a + b) (a - b)

= (x² + 9) (x² - 9)

= (x² + 9) (x + 3) (x - 3)

Problem 7 :

x⁴ – 16

Solution :

= x⁴ – 16

(a² - b²) = (a + b) (a - b)

= (x² + 4) (x² - 4)

= (x² + 4) (x + 2) (x - 2)

Problem 8 :

2x³ + 13x² + 15x

Solution :

= 2x³ + 13x² + 15x

= 2x³ + 10x² + 3x² + 15x

= 2x²(x + 5) + 3x(x + 5)

= (2x² + 3x) (x + 5)

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

Problem 9 :

4x³ - 10x² + 6x

Solution :

= 4x³ - 10x² + 6x

= 4x³ - 4x² - 6x² + 6x

= 4x²(x - 1) – 6x(x - 1)

= (4x² - 6x) (x - 1)

= 2x(2x – 3) (x - 1)

Problem 10 :

z⁴ – 12z² + 27

Solution :

= z⁴ – 12z² + 27

= z⁴ – 3z² - 9z² + 27

= z²(z² - 3) - 9(z² - 3)

= (z² - 9) (z² - 3)

= (z + 3) (z - 3) (z² - 3)

Problem 11 :

(c + 2)² - 1

Solution :

= (c + 2)² - 1

= c² + 4c + 4 – 1

= c² + 4c + 3

= c² + c + 3c + 3

= c(c + 1) + 3(c + 1)

= (c + 3) (c + 1)

Problem 12 :

4 – (y - 1)²

Solution :

= 4 – (y - 1)²

= 2² - (y - 1)²

Using the algebraic identity, we get

= (2 + (y - 1))(2 - (y - 1))

= (2 + y - 1) (2 - y + 1)

= (y + 1) (3 - y)

Problem 13 :

x²y – 16y

Solution :

= x²y – 16y

= y(x² - 16)

(a² - b²) = (a + b) (a - b)

= y(x + 4) (x - 4)

Problem 14 :

3(x - 1)² - 12

Solution:

= 3(x - 1)² - 12

= 3(x² + 1 – 2x) – 12

= 3x² + 3 – 6x – 12

= 3x² - 6x – 9

= 3x² + 3x – 9x – 9

= 3x(x + 1) – 9(x + 1)

= (3x - 9) (x + 1)

= 3(x - 3) (x + 1) 

Problem 15 :

9 – 9(x + 2)²

Solution :

= 9 – 9(x + 2)²

Factoring 9, we get

= 9[1 – (x + 2)²]

= 9 [1² – (x + 2)²]

= 9 [(1+ x + 2) (1 - x - 2)]

= -9 [(x + 3) (x + 1)]