Certainly! Let's perform the polynomial division step-by-step.
We are given the polynomial [tex]\( x^4 + 14 \)[/tex] and we need to divide it by [tex]\( x + 2 \)[/tex].
### Step 1: Setup the Division
We set up the polynomial division in a long division format:
[tex]\[ \frac{x^4 + 14}{x + 2} \][/tex]
### Step 2: Divide the Leading Term
First, we divide the leading term of the numerator ([tex]\( x^4 \)[/tex]) by the leading term of the divisor ([tex]\( x \)[/tex]):
[tex]\[ \frac{x^4}{x} = x^3 \][/tex]
This gives us the first term of the quotient:
[tex]\[ x^3 \][/tex]
### Step 3: Multiply and Subtract
Next, we multiply [tex]\( x^3 \)[/tex] by [tex]\( x + 2 \)[/tex]:
[tex]\[ x^3 \cdot (x + 2) = x^4 + 2x^3 \][/tex]
We then subtract this result from the original polynomial:
[tex]\[ (x^4 + 14) - (x^4 + 2x^3) = -2x^3 + 14 \][/tex]
### Step 4: Repeat the Process
We repeat the process with the new polynomial [tex]\( -2x^3 + 14 \)[/tex].
#### Divide the leading term:
[tex]\[ \frac{-2x^3}{x} = -2x^2 \][/tex]
#### Multiply and subtract:
[tex]\[ -2x^2 \cdot (x + 2) = -2x^3 - 4x^2 \][/tex]
[tex]\[ (-2x^3 + 14) - (-2x^3 - 4x^2) = 4x^2 + 14 \][/tex]
### Step 5: Continue the Process
We continue with the new polynomial [tex]\( 4x^2 + 14 \)[/tex].
#### Divide the leading term:
[tex]\[ \frac{4x^2}{x} = 4x \][/tex]
#### Multiply and subtract:
[tex]\[ 4x \cdot (x + 2) = 4x^2 + 8x \][/tex]
[tex]\[ (4x^2 + 14) - (4x^2 + 8x) = -8x + 14 \][/tex]
### Step 6: Repeat Until We Reach the Constant Term
We proceed with [tex]\( -8x + 14 \)[/tex].
#### Divide the leading term:
[tex]\[ \frac{-8x}{x} = -8 \][/tex]
#### Multiply and subtract:
[tex]\[ -8 \cdot (x + 2) = -8x - 16 \][/tex]
[tex]\[ (-8x + 14) - (-8x - 16) = 30 \][/tex]
### Step 7: Assemble the Quotient and Remainder
After completing the division, we have:
- Quotient: [tex]\( x^3 - 2x^2 + 4x - 8 \)[/tex]
- Remainder: [tex]\( 30 \)[/tex]
Therefore, the polynomial division yields:
[tex]\[ x^3 - 2x^2 + 4x - 8 + \frac{30}{x + 2} \][/tex]
### Final Answer
So, we filled the quotients and the remainder as:
[tex]\[ x^3 - 2x^2 + 4x - 8 + \frac{30}{x + 2} \][/tex]
