To determine the value of [tex]\( m \)[/tex] for which the remainder is [tex]\(-5\)[/tex] when the polynomial [tex]\( 3x^2 + mx - 2 \)[/tex] is divided by [tex]\( x + 2 \)[/tex], we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial [tex]\( f(x) \)[/tex] by a linear divisor [tex]\( x - c \)[/tex] is [tex]\( f(c) \)[/tex].
Here, the divisor is [tex]\( x + 2 \)[/tex], which can be written as [tex]\( x - (-2) \)[/tex]. Therefore, we need to find the value of the polynomial [tex]\( 3x^2 + mx - 2 \)[/tex] when [tex]\( x = -2 \)[/tex].
Given polynomial:
[tex]\[ f(x) = 3x^2 + mx - 2 \][/tex]
We substitute [tex]\( x = -2 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(-2) = 3(-2)^2 + m(-2) - 2 \][/tex]
Now, let's calculate each term step by step:
1. Calculate [tex]\( 3(-2)^2 \)[/tex]:
[tex]\[ 3(-2)^2 = 3 \cdot 4 = 12 \][/tex]
2. Calculate [tex]\( m(-2) \)[/tex]:
[tex]\[ m(-2) = -2m \][/tex]
3. Combine these results with the constant term [tex]\( -2 \)[/tex]:
[tex]\[ f(-2) = 12 - 2m - 2 \][/tex]
Simplify the expression:
[tex]\[ f(-2) = 12 - 2 - 2m \][/tex]
[tex]\[ f(-2) = 10 - 2m \][/tex]
According to the problem, the remainder when the polynomial [tex]\( 3x^2 + mx - 2 \)[/tex] is divided by [tex]\( x + 2 \)[/tex] is [tex]\(-5\)[/tex]. Thus, we set the polynomial evaluated at [tex]\( x = -2 \)[/tex] equal to the remainder:
[tex]\[ 10 - 2m = -5 \][/tex]
Now, solve for [tex]\( m \)[/tex]:
1. Subtract 10 from both sides:
[tex]\[ 10 - 2m - 10 = -5 - 10 \][/tex]
[tex]\[ -2m = -15 \][/tex]
2. Divide both sides by -2:
[tex]\[ m = \frac{-15}{-2} \][/tex]
[tex]\[ m = \frac{15}{2} \][/tex]
Therefore, the value of [tex]\( m \)[/tex] is:
[tex]\[ \boxed{\frac{15}{2}} \][/tex]
