Answer :
Final answer:
The remainder when f(x) is divided by (x + 3) is 22. The polynomial functions with (x + 3) as a factor are f(x) = x^4 - 12x^3 + 54x^2 - 108x + 81 and f(x) = x^5 + 2x^4 - 23x^3 - 60x^2. The first line in which an error occurred in testing x = 1 as a zero for f(x) = x^3 - 3x + 2 is the third line. The real zeros of f(x) = 3x^3 + 11x^2 + 8x - 4 are x = -4/3 and x = 1/3.
Explanation:
To find the remainder when the function f(x) = x^3 - 2x^2 - 5x + 16 is divided by (x + 3), we can use synthetic division. Dividing f(x) by (x + 3) gives us a remainder of 22. Therefore, the remainder when f(x) is divided by (x + 3) is 22. The correct option is d. 22.
To determine which polynomial functions have (x + 3) as a factor, we can use synthetic division. By dividing each function by (x + 3), we find that f(x) = x^4 - 12x^3 + 54x^2 - 108x + 81 and f(x) = x^5 + 2x^4 - 23x^3 - 60x^2 have (x + 3) as a factor. Therefore, the correct options are a. and c.
To test whether x = 1 is a zero of the function f(x) = x^3 - 3x + 2, we substitute x = 1 into the function and evaluate it. The error in the steps provided occurs in the third line, where the constant term (2) is incorrectly subtracted from the function value (-4), resulting in an incorrect value of -2. Therefore, the first line in which an error occurred is the third line.
Using synthetic division, we can find the real zeros of the function f(x) = 3x^3 + 11x^2 + 8x - 4. By dividing f(x) by the potential zeros, we find that the real zeros are x = -4/3 and x = 1/3.
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