Divide the third-degree polynomial, 8x3+20x2−12x%0D%0A8%0D%0A𝑥%0D%0A3%0D%0A+%0D%0A20%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A12%0D%0A𝑥%0D%0A, by the first-degree polynomial, x+3%0D%0A𝑥%0D%0A+%0D%0A3%0D%0A, using long division. What is the result?%0D%0A%0D%0AOption #1: 8x2+44x+−144x+3%0D%0A8%0D%0A𝑥%0D%0A2%0D%0A+%0D%0A44%0D%0A𝑥%0D%0A+%0D%0A−%0D%0A144%0D%0A𝑥%0D%0A+%0D%0A3%0D%0A%0D%0AOption #2: 8x2−4x%0D%0A8%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A4%0D%0A𝑥%0D%0A%0D%0AOption #3: 8x2−4x+12xx+3%0D%0A8%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A4%0D%0A𝑥%0D%0A+%0D%0A12%0D%0A𝑥%0D%0A𝑥%0D%0A+%0D%0A3%0D%0A%0D%0AOption #4: 8x2+44x+120xx+3

Answer 1

To divide the polynomial \(8x^3 + 20x^2 - 12x\) by \(x + 3\) using long division, follow these steps:

1. Divide the leading term of the dividend by the leading term of the divisor: \[ \frac{8x^3}{x} = 8x^2 \]

2. Multiply \(8x^2\) by the entire divisor \(x + 3\): \[ 8x^2 \cdot (x + 3) = 8x^3 + 24x^2 \]

3. Subtract this result from the original polynomial: \[ (8x^3 + 20x^2 - 12x) - (8x^3 + 24x^2) = -4x^2 - 12x \]

4. Bring down any remaining terms (in this case, none), and repeat the division:

   • Now divide the leading term \(-4x^2\) by \(x\): \[ \frac{-4x^2}{x} = -4x \]

5. Multiply \(-4x\) by \(x + 3\): \[ -4x \cdot (x + 3) = -4x^2 - 12x \]

6. Subtract this from \(-4x^2 - 12x\): \[ (-4x^2 - 12x) - (-4x^2 - 12x) = 0 \]

Since the remainder is 0, we can conclude that the polynomial \(8x^3 + 20x^2 - 12x\) divides perfectly by \(x + 3\).

The quotient of the division is: \[ 8x^2 - 4x \]

Now, comparing this result to your options:

• Option #1: \(8x^2 + 44x - 144\) → Not correct
• Option #2: \(8x^2 - 4x\) → Correct!
• Option #3: \(8x^2 - 4x + \frac{12}{x+3}\) → Not appropriate
• Option #4: \(8x^2 + 44x + \frac{120}{x+3}\) → Not correct

Thus, the correct answer is Option #2: \(8x^2 - 4x\).