To divide the polynomial \(2x^3 + 4x^2 + 5x - 6\) by \(x + 1\) using synthetic division, we first rewrite the divisor in a way that facilitates synthetic division. We can use \(c = -1\) (since \(x + 1 = 0\) leads to \(x = -1\)).
The coefficients of the dividend \(2x^3 + 4x^2 + 5x - 6\) are \(2\), \(4\), \(5\), and \(-6\).
Setting up synthetic division:
- Write the coefficients: \(2, 4, 5, -6\).
- Use \(-1\) as the number to perform synthetic division.
Now, perform synthetic division:
-1 | 2 4 5 -6
| -2 -2 -3
---------------------
2 2 3 -9
Here's what happens step-by-step:
- Bring down the first coefficient \(2\).
- Multiply \(-1\) by \(2\) and write it under the next coefficient (4): \( -1 \cdot 2 = -2\).
- Add \(4 + (-2) = 2\).
- Multiply \(-1\) by \(2\) and write it under the next coefficient (5): \( -1 \cdot 2 = -2\).
- Add \(5 + (-2) = 3\).
- Multiply \(-1\) by \(3\) and write it under the next coefficient (-6): \( -1 \cdot 3 = -3\).
- Add \(-6 + (-3) = -9\).
The synthetic division gives us the remainder \(-9\) and the coefficients of the quotient polynomial as \(2\), \(2\), and \(3\).
The quotient is therefore:
\[ 2x^2 + 2x + 3 \]
Since we are asked for a missing term in the quotient, it would be more accurate to note that there is no missing term in the polynomial, but if you were searching for a particular coefficient, \(2, 2, 3\) correspond to \(x^2, x, \) and the constant term respectively.
So the final answer is:
Quotient: \(2x^2 + 2x + 3\)
We have no missing term since all coefficients are accounted for.