Use long division to divide the polynomial 15x2+14x−8 by 3x+4 . What is the quotient?(1 point)

Answer 1

To divide the polynomial \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) using long division, we will follow these steps:

1. Setup the division: We write \( 15x^2 + 14x - 8 \) (the dividend) under the long division symbol and \( 3x + 4 \) (the divisor) outside.

2. Divide the leading term: We start by dividing the leading term of the dividend \( 15x^2 \) by the leading term of the divisor \( 3x \).

   \[ \frac{15x^2}{3x} = 5x \]

3. Multiply and subtract: Next, we multiply \( 5x \) by the entire divisor \( 3x + 4 \):

   \[ 5x(3x + 4) = 15x^2 + 20x \]

   Now we subtract \( (15x^2 + 20x) \) from the original polynomial:

   \[ (15x^2 + 14x - 8) - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8 \]

4. Bring down the next term: Since we only have the remainder \( -6x - 8 \) at this point, we don't need to bring down anything (as there are no remaining terms to bring down).

5. Repeat the process: Now, we divide the new leading term \( -6x \) by \( 3x \):

   \[ \frac{-6x}{3x} = -2 \]

6. Multiply and subtract again: We multiply \( -2 \) by the entire divisor:

   \[ -2(3x + 4) = -6x - 8 \]

   Now, subtract \( (-6x - 8) \) from the current remainder \( -6x - 8 \):

   \[ (-6x - 8) - (-6x - 8) = 0 \]

7. Conclusion: There is no remainder, and the division is complete. Our final quotient is:

   \[ \boxed{5x - 2} \]