Question

One factor of f (x ) = 4 x cubed minus 4 x squared minus 16 x + 16 is (x – 2). What are all the roots of the function? Use the Remainder Theorem.

Answer 1

$f(x) =4 x {}^{3} - 4x {}^{2} - 16x + 16$

$divide \: by \: x - 2$

$4x {}^{2} \: into \: (x - 2) = 4x {}^{3} - 8x {}^{2}$

$g(x) = 4x {}^{2} - 16x + 16$

$4x \: into \: (x - 2) = 4x {}^{2} - 8x$

$z(x) = - 8x + 16$

$- 8 \: into \: (x - 2) = - 8x + 16 \\ no \: remainder$

$f(x) = (x - 2)(4x {}^{2} + 4x - 8)$

$f(x) = 0 \\ x - 2 = 0 \: \: \: \: \: \: \: \: \: 4x {}^{2} + 4x - 8 = 0$

$x = \frac{ - 4 + \sqrt{4 {}^{2} - 4(4)( - 8) } }{2(4)} = \frac{ - 4 + \sqrt{144} }{8} = \frac{ - 4 + 12}{8} = 1$

$x = 2$

$x = \frac{ - 4 - 12}{8} = \frac{ - 16}{8} = - 2$