Question

Use synthetic division to decide whether the given number k is a zero of the given polynomial function. If it is not, give the value of f(k).

f(x) = x^2 + 2x^2 - 3x + 1, k = 2 + i

A) The given k is not a zero of the polynomial function, f(2 + i) = 4 + 5i
B) The given k is a zero of the polynomial function.

Answer 1

The given number k = 2 + i is not a zero of the polynomial function f(x). The value of f(2 + i) is i.

Step-by-step explanation:

To determine whether the given number k = 2 + i is a zero of the polynomial function f(x) = x^2 + 2x^2 - 3x + 1, we can use synthetic division.

1. Set up the synthetic division by putting the coefficients of the polynomial in the top row, and then write the value of k to the left of the vertical line.
2. Perform the synthetic division. Multiply the value of k by the leading coefficient and write the result below the second coefficient. Add the two values together and write the sum below the third coefficient. Continue this process until you reach the constant term.
3. If the remainder is zero, then k is a zero of the polynomial function. If the remainder is not zero, then f(k) gives the value of the polynomial function at k.

In this case, when performing the synthetic division, we find that the remainder is not zero. Therefore, the given k = 2 + i is not a zero of the polynomial function f(x). The value of f(2 + i) can be determined by substituting the value of k into the polynomial function: f(2 + i) = (2 + i)^2 + 2(2 + i)^2 - 3(2 + i) + 1 = 4 + 4i + i^2 + 8 + 8i + 2i^2 - 6 - 3i + 1 = (4 + 2i - 2) + (4i - 3i) + (i^2 + 1) = 0 + i + 0 = i.