Sure, we need to perform synthetic division on the polynomials F(X) = 2x^4 + 3x^2 + 1 and G(X) = x + 1. Using the synthetic division algorithm, here are the steps we will follow:
1. Write the coefficients of the polynomials F(X) as:
* For F(X), the list of coefficients is: [2, 0, 3, 0, 1].
* Here, we need to include zeros for the missing powers of x in F(x). Hence, the coefficient for x^3 and x are 0.
2. Write the root of the polynomial G(X). The root is the value of x that makes the polynomial G(X) equal to zero. For our polynomial, G(X) = X + 1, x = -1 is the root since (-1) + 1 equals 0.
3. Now, it's time to perform synthetic division. Bring down the leading coefficient of polynomial F(X), which is 2.
4. Multiply this number by the root of polynomial G(X), which is -1. Write the result underneath the second coefficient and then add them.
5. Continue this process: multiply the number just found in the previous step by the root and write underneath the coefficient in the next line.
6. You continue this process until every coefficient from the F polynomial has been exhausted.
After applying these steps, we found that the coefficients of the quotient function are: [2, -2, 5, -5]. This equates to polynomial 2x^3 - 2x^2 + 5x - 5.
Lastly, check the last number that you obtained (which is 6). This is the remainder of the division.
Therefore, we can conclude that the result of the division is:
F(X) = G(X) * quotient + remainder
By substituting the values, it becomes:
2x^4 + 3x^2 + 1 = (x+1) * (2x^3 - 2x^2 + 5x - 5) + 6.