Certainly! Let's solve this step-by-step.
Given the polynomial:
[tex]\[ p(x) = x^3 - (m-2)x^2 - mx + 28 \][/tex]
We know that when this polynomial is divided by [tex]\( (x+3) \)[/tex], the remainder is 10. By the Remainder Theorem, the remainder when [tex]\( p(x) \)[/tex] is divided by [tex]\( (x+3) \)[/tex] is [tex]\( p(-3) \)[/tex].
So, let's plug [tex]\( x = -3 \)[/tex] into the polynomial and set it equal to 10:
[tex]\[ p(-3) = (-3)^3 - (m-2)(-3)^2 - m(-3) + 28 = 10 \][/tex]
Now, let's evaluate each term separately:
1. [tex]\( (-3)^3 = -27 \)[/tex]
2. [tex]\( (-3)^2 = 9 \)[/tex]
3. So, [tex]\( -(m-2)(-3)^2 = -(m-2) \cdot 9 = -9(m-2) \)[/tex]
4. And, [tex]\( -m(-3) = 3m \)[/tex]
Substituting these into the polynomial:
[tex]\[ -27 - 9(m-2) + 3m + 28 = 10 \][/tex]
Now, let's simplify the equation:
[tex]\[ -27 - 9m + 18 + 3m + 28 = 10 \][/tex]
Combine the like terms:
[tex]\[ -27 + 18 + 28 - 9m + 3m = 10 \][/tex]
[tex]\[ 19 - 6m = 10 \][/tex]
To isolate [tex]\( m \)[/tex], we solve the equation:
[tex]\[ 19 - 6m = 10 \][/tex]
[tex]\[ -6m = 10 - 19 \][/tex]
[tex]\[ -6m = -9 \][/tex]
Now, divide both sides by -6:
[tex]\[ m = \frac{-9}{-6} \][/tex]
[tex]\[ m = \frac{3}{2} \][/tex]
Therefore, the value of [tex]\( m \)[/tex] is:
[tex]\[ m = 1.5 \][/tex]
