Synthetic division is a shorthand method of polynomial division, particularly useful when dividing by a linear factor. To work through the problem of dividing the polynomial x4 + x + 1 by x3 + 2, we first set up our synthetic division.
Since we are dividing by x3 + 2, we can rearrange this into a form suitable for synthetic division. However, since this divisor is a cubic polynomial, synthetic division isn’t directly applicable in the traditional sense. Typically, synthetic division is used with linear divisors (like x – c).
Instead, we can perform polynomial long division, or find the roots of the divisor if applicable, and then apply synthetic division accordingly. For this example, let’s illustrate a manual polynomial division:
1. Write out the polynomials: - Dividend: x4 + 0x3 + 0x2 + x + 1 - Divisor: x3 + 0x2 + 0x + 2 2. Divide the leading term of the dividend by the leading term of the divisor: - x4 ÷ x3 = x 3. Multiply the entire divisor by x: - (x3 + 2) * x = x4 + 2x 4. Subtract this result from the original polynomial: - (x4 + 0x3 + 0x2 + x + 1) - (x4 + 2x) = -2x + 1 5. Now drop down the remaining terms if necessary. In this case, there are no additional terms from the dividend since we've factored out the leading terms.
Thus, we can see that our polynomial can’t be reduced further using synthetic division in this scenario directly, as we can’t reach a zero remainder unless we find more roots of the divisor.
Finally, the quotient from our division process indicates that the answer, when divide x4 + x + 1 by x3 + 2, results in:
- Quotient: x
- Remainder: -2x + 1
In summary, the division of x4 + x + 1 by x3 + 2 yields a quotient of x and a remainder of -2x + 1.