Certainly! Let's tackle the problem step by step.
### Step 1: Find the remainder when [tex]\( f(x) = 3x^3 + x^2 + x - 4 \)[/tex] is divided by [tex]\( x + 1 \)[/tex].
To find the remainder when a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 1 \)[/tex], we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of this division is [tex]\( f(-1) \)[/tex].
So, let's evaluate [tex]\( f(-1) \)[/tex]:
[tex]\[
f(x) = 3x^3 + x^2 + x - 4
\][/tex]
Now plug in [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 3(-1)^3 + (-1)^2 + (-1) - 4
= 3(-1) + 1 - 1 - 4
= -3 + 1 - 1 - 4
= -7
\][/tex]
Therefore, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 1 \)[/tex] is [tex]\( -7 \)[/tex].
### Step 2: Find the value of [tex]\( k \)[/tex] for which [tex]\( g(x) = f(x) + k \)[/tex] is divisible by [tex]\( x + 1 \)[/tex].
For [tex]\( g(x) \)[/tex] to be divisible by [tex]\( x + 1 \)[/tex], the remainder when dividing [tex]\( g(x) \)[/tex] by [tex]\( x + 1 \)[/tex] must be 0.
Since [tex]\( g(x) = f(x) + k \)[/tex], we can express the division as:
[tex]\[
g(x) = (x + 1)Q(x) + R
\][/tex]
Here, [tex]\( R \)[/tex] is the remainder, and it must equal 0 for [tex]\( g(x) \)[/tex] to be divisible by [tex]\( x + 1 \)[/tex].
We already know that [tex]\( f(x) \)[/tex] leaves a remainder of [tex]\( -7 \)[/tex]:
[tex]\[
f(x) = (x + 1)Q(x) - 7
\][/tex]
Thus,
[tex]\[
g(x) = f(x) + k = (x + 1)Q(x) - 7 + k
\][/tex]
For [tex]\( g(x) \)[/tex] to be divisible by [tex]\( x + 1 \)[/tex]:
[tex]\[
-7 + k = 0 \Rightarrow k = 7
\][/tex]
Therefore, [tex]\( k \)[/tex] must be 7.
### Step 3: Find the other factors of [tex]\( g(x) \)[/tex].
Now we have [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = f(x) + k = 3x^3 + x^2 + x - 4 + 7 = 3x^3 + x^2 + x + 3
\][/tex]
We can factorize [tex]\( g(x) \)[/tex]. Since [tex]\( g(x) \)[/tex] is divisible by [tex]\( x + 1 \)[/tex], we can write:
[tex]\[
g(x) = (x + 1)Q(x)
\][/tex]
To determine [tex]\( Q(x) \)[/tex], we perform polynomial division of [tex]\( g(x) \)[/tex] by [tex]\( x + 1 \)[/tex]:
Dividing [tex]\( 3x^3 + x^2 + x + 3 \)[/tex] by [tex]\( x + 1 \)[/tex], we get:
[tex]\[
g(x) = (x + 1)(3x^2 - 2x + 3)
\][/tex]
Thus, the complete factorization of [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = (x + 1)(3x^2 - 2x + 3)
\][/tex]
### Summary:
1. The remainder when [tex]\( f(x) = 3x^3 + x^2 + x - 4 \)[/tex] is divided by [tex]\( x + 1 \)[/tex] is [tex]\( -7 \)[/tex].
2. The value of [tex]\( k \)[/tex] for which [tex]\( g(x) = f(x) + k \)[/tex] is divisible by [tex]\( x + 1 \)[/tex] is [tex]\( k = 7 \)[/tex].
3. The other factor of [tex]\( g(x) = 3x^3 + x^2 + x + 3 \)[/tex] is [tex]\( 3x^2 - 2x + 3 \)[/tex].
Thus, the factorized form of [tex]\( g(x) \)[/tex] is [tex]\( g(x) = (x + 1)(3x^2 - 2x + 3) \)[/tex].
