What is the quotient in synthetic division?
Understand the Problem
The question is asking for the result (quotient) obtained when performing synthetic division, which is a method used to divide polynomials. To solve this, we would perform synthetic division with a specified polynomial and divisor.
Answer
The quotient is $2x^2 + 9x + 22$ with a remainder of $72$.
Answer for screen readers
The quotient is $2x^2 + 9x + 22$ with a remainder of $72$.
Steps to Solve
- Identify the dividend and divisor
For synthetic division, we need to have the polynomial (the dividend) and the value we are dividing by (the divisor). For example, if we are dividing the polynomial $2x^3 + 3x^2 - 5x + 6$ by $x - 3$, we set up the synthetic division with 3 as the divisor.
- Set up the synthetic division
Write down the coefficients of the polynomial in order. For $2x^3 + 3x^2 - 5x + 6$, the coefficients are 2, 3, -5, and 6. We set it up as follows:
3 | 2 3 -5 6
- Perform the synthetic division
Bring down the leading coefficient (2) directly beneath the line. Next, multiply the divisor (3) by this number (2) and write the result (6) under the next coefficient (3).
3 | 2 3 -5 6
| 6
----------------
2 9
Then, continue this process of multiplying and adding down the line. Multiply 3 by 9 (the new number) and write it under -5:
3 | 2 3 -5 6
| 6 27
----------------
2 9 22
Finally, multiply 3 by 22 and add to 6:
3 | 2 3 -5 6
| 6 27 66
----------------
2 9 22 72
- Write the result
The final row gives us the coefficients of the quotient polynomial. The numbers represent the coefficients of 2, 9, and 22, which corresponds to the polynomial:
$$2x^2 + 9x + 22$$
The remainder from this division is 72.
- State the final result
Thus, the result of the synthetic division is that the quotient is $2x^2 + 9x + 22$ and the remainder is 72.
The quotient is $2x^2 + 9x + 22$ with a remainder of $72$.
More Information
Synthetic division is a quick way to divide polynomials, particularly useful for linear denominators. The quotient gives us a new polynomial, while the remainder indicates what was left over from the division.
Tips
- Forgetting to bring down the leading coefficient.
- Mixing up signs when subtracting the multiplied values.
- Not writing the final result as a polynomial correctly, especially when including the remainder.
