Question

Prove that (5x + 4) is a factor of 5x³ + 4x² – 5x – 4. Hence factorize the given polynomial completely.

Answer 1

f(x) = 5x³ + 4x² – 5x – 4
Let 5x + 4 = 0, then 5x = -4
⇒ x = `(-4)/(5)`

∴ `f(-4/5) = 5(-4/5)^3 + 4(-4/5)^2 -5(-4/5) -4`

= `5 xx (-64/125) + 4 xx (16)/(25) + 4 - 4`

= `-(64)/(25) + (64)/(25) + 4 - 4` = 0

∵ `f(-4/5)` = 0
∴ (5x + 4) is a factor of f(x)
Now dividing f(x) by 5x + 4, we get
5x³ + 4x² – 5x – 4
= (5x + 4)(x² – 1)
= (5x + 4)[(x)² – (1)²]
= (5x + 4)(x + 1)(x – 1)

`5x + 4")"overline(5x^3 + 4x^2 - 5x - 4)("x^2 - 1`
             5x³ + 4x²
            –      –                          
                    –5x – 4
                    –5x – 4
                   +     +     
                         x