Question

Find the value of p so that 3x-1 is a factor of 8x^4 + 4x^3 - 16x^2 + 10x + p.

Answer 1

given that 3x-1 is a factor of 8x^4 + 4x^3 - 16x^2 + 10x + p.

Equate 3x-1 to 0.
3x - 1 = 0
⇒ 3x = 1
⇒ x = 1/3

Since 3x-1 is a factor of 8x^4 + 4x^3 - 16x^2 + 10x + p, x=1/3 is a solution of 
8x^4 + 4x^3 - 16x^2 + 10x + p = 0.

So put x=1/3 in the equation 8x^4 + 4x^3 - 16x^2 + 10x + p = 0.

$8x^4 + 4x^3 - 16x^2 + 10x + p=0\\ \\ \Rightarrow 8( \frac{1}{3} )^4 + 4( \frac{1}{3} )^3 - 16( \frac{1}{3} )^2 + 10( \frac{1}{3} ) + p = 0\\ \\ \Rightarrow 8( \frac{1}{81} ) + 4( \frac{1}{27} ) - 16( \frac{1}{9} ) + 10( \frac{1}{3} ) + p = 0\\ \\ \Rightarrow \frac{8}{81}+ \frac{4}{27}-\frac{16}{9} +\frac{10}{3}+ p = 0\\ \\ \Rightarrow \frac{8+12-144+270}{81}+p=0 \\ \\ \Rightarrow \frac{146}{81}+p=0 \\ \\ \Rightarrow p=-\frac{146}{81}$

Answer 2

If 3x-1 is a factor then 3x = 1 
⇒x = 1/3
Now , 8x⁴+4x³-16x²+10x+p = 0 
       ⇒8(1/3)⁴+4(1/3)³-16(1/3)²+10(1/3) = -p
       ⇒8(1/81)+4(1/27)-16(1/9)+10/3 = -p
       ⇒8/81+4/27-16/9+10/3 = -p
       ⇒(8+12-144+270)/81 = -p
       ⇒146/81 = -p
Therefore p = -146/81 .