# Question #55c5d

Sep 5, 2017

There are two possible values:

$m = - \frac{3}{2}$ or $m = 2$

#### Explanation:

We use the remainder theorem:

The remainder of the division of a polynomial $f \left(x\right)$ by a linear factor $x - a$ is $f \left(a\right)$

Consider the first polynomial:

$P \left(x\right) = {x}^{3} + 4 {x}^{2} - 2 x + 1$

Then by the Remainder Theorem, if we divide $P \left(x\right)$ by $x - m$ then the remainder, ${r}_{p}$, is:

${r}_{p} = P \left(m\right) = {m}^{3} + 4 {m}^{2} - 2 m + 1$

Similarly for the second polynomial:

$Q \left(x\right) = {x}^{3} + 2 {x}^{2} - x + 7$

If we divide $Q \left(x\right)$ by $x - m$ then the remainder, ${r}_{q}$, is:

${r}_{q} = Q \left(m\right) = {m}^{3} + 2 {m}^{2} - m + 7$

We are given that the remainders are the same:

$\therefore {r}_{p} = {r}_{q}$
$\therefore {m}^{3} + 4 {m}^{2} - 2 m + 1 = {m}^{3} + 2 {m}^{2} - m + 7$
$\therefore 2 {m}^{2} - m - 6 = 0$
$\therefore \left(m - 2\right) \left(2 m + 3\right) = 0$
$\therefore m = - \frac{3}{2} , 2$