Question

Without actual division, prove that 3x⁴- 2x³ - 2x² - 2x - 5 is exactly divisible
by 3x²- 2x -5.​

Answer 1

Step-by-step explanation:

First, we factorize 3x²- 2x -5

3x²- 2x -5

= 3x²- 5x + 3x -5

= x (3x - 5) + 1 (3x -5)

= (x+ 1)(3x - 5)

Second, we use (x + 1) and (3x -5) in 3x⁴- 2x³ - 2x² - 2x - 5

x + 1 = 0

x = -1

p(-1) = 3(-1)⁴- 2(-1)³ - 2(-1)² - 2(-1) - 5

= 3 + 2 - 2 + 2 - 5

= 0

3x - 5 =0

x = 5/3

p(5/3) = 3(5/3)⁴- 2(5/3)³ - 2(5/3)² - 2(5/3) - 5

= 625/27 - 250/27 - 50/9 - 10/3 - 5

= 625/27 - 250/27 - 150/27 - 90/27 - 135/27

= 0/27

=0

Therefore

3x⁴- 2x³ - 2x² - 2x - 5 is exactly divisible

by 3x²- 2x -5