Using Synthetic Division What is the Quotient of 2x³ + 3x + 10x²?

To perform synthetic division on the polynomial 2x³ + 10x² + 3x, we must first identify what we are dividing by. For this example, let’s assume we are dividing by (x – 1). The first step is to write down the coefficients of the polynomial:

• 2 (for x³)
• 10 (for x²)
• 3 (for x)
• 0 (for the constant term, since there is no constant)

Now, we set up our synthetic division:

1 | 2   10   3   0
  |      2  12  15
-------------------
  2   12  15  15

Here’s a step-by-step explanation of the process:

1. Bring down the leading coefficient (2) straight down.
2. Multiply it by the divisor’s root (1) and add it to the next coefficient (10): 1 * 2 + 10 = 12.
3. Repeat this process: multiply 12 by 1 and add to 3: 1 * 12 + 3 = 15.
4. Finally, multiply 15 by 1 and add to 0: 1 * 15 + 0 = 15.

The result of our synthetic division shows that the quotient is 2x² + 12x + 15 and the remainder is 15. Therefore, the final result of dividing 2x³ + 10x² + 3x by (x – 1) is:

Quotient: 2x² + 12x + 15
Remainder: 15