To solve the problem of finding the quotient and the remainder when [tex]\(x^4 - 3x^2 + 21x + 9\)[/tex] is divided by [tex]\(x + 3\)[/tex], we can use polynomial long division. Let me guide you through the process step by step:
### Step 1: Setup the Division
We want to divide the polynomial:
[tex]\[ x^4 - 3x^2 + 21x + 9 \][/tex]
by the divisor:
[tex]\[ x + 3 \][/tex]
### Step 2: Perform Polynomial Long Division
#### a. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{x^4}{x} = x^3 \][/tex]
#### b. Multiply the entire divisor by this result:
[tex]\[ x^3 \cdot (x + 3) = x^4 + 3x^3 \][/tex]
#### c. Subtract this from the original polynomial:
[tex]\[ (x^4 - 3x^2 + 21x + 9) - (x^4 + 3x^3) = -3x^3 - 3x^2 + 21x + 9 \][/tex]
#### d. Repeat the process with the new polynomial:
[tex]\[ \frac{-3x^3}{x} = -3x^2 \][/tex]
[tex]\[ -3x^2 \cdot (x + 3) = -3x^3 - 9x^2 \][/tex]
[tex]\[ (-3x^3 - 3x^2 + 21x + 9) - (-3x^3 - 9x^2) = 6x^2 + 21x + 9 \][/tex]
#### e. Continue with the new result:
[tex]\[ \frac{6x^2}{x} = 6x \][/tex]
[tex]\[ 6x \cdot (x + 3) = 6x^2 + 18x \][/tex]
[tex]\[ (6x^2 + 21x + 9) - (6x^2 + 18x) = 3x + 9 \][/tex]
#### f. Finally:
[tex]\[ \frac{3x}{x} = 3 \][/tex]
[tex]\[ 3 \cdot (x + 3) = 3x + 9 \][/tex]
[tex]\[ (3x + 9) - (3x + 9) = 0 \][/tex]
### Step 3: Collect Results
From the polynomial long division steps, we see that the quotient is:
[tex]\[ x^3 - 3x^2 + 6x + 3 \][/tex]
and the remainder is:
[tex]\[ 0 \][/tex]
### Step 4: Write the Final Answer in the Desired Form
The final quotient and remainder form is:
[tex]\[ \frac{x^4 - 3x^2 + 21x + 9}{x + 3} = (x^3 - 3x^2 + 6x + 3) + \frac{0}{x + 3} \][/tex]
Hence, the final answer is:
[tex]\[ \boxed{x^3 - 3x^2 + 6x + 3} \][/tex]
