Question

Select the correct answer from each drop-down menu. Use the remainder theorem to verify this statement. (x + 5) is a factor of the function f(x) = x3 + 3x2 – 25x – 75

1.) Find the ------------------ of f(x) =x^3 + 3x^2 - 25x - 75
2.) The --------------- of this operation is 0
3.) Therefore, (x+5) is --------------- of function F.
4.) So f(---------------------) = 0
Options for 1.) Difference, Product, Quotient, Sum
Options for 2.) Discriminant, Remainder, Sum, Quotient
Options for 3.) The simplified form, The opposite, A factor, Not a factor
Options for 4.) -5, 75, -75, 5

Answer 1

To determine if (x + 5) is a factor of f(x), we substitute x with -5 and find the sum. If the remainder of f(-5) is 0, then (x + 5) is confirmed as a factor of f(x).

Step-by-step explanation:

To verify whether (x + 5) is a factor of the function f(x) = x3 + 3x2 − 25x − 75 using the remainder theorem, we must proceed through the following steps:

1. Find the sum of f(x) by substituting x with the value that makes (x + 5) = 0. In this case, that value is x = -5.
2. The remainder of this operation will reveal if (x + 5) is a factor of f(x). If the remainder is 0, then (x + 5) is a factor.
3. Therefore, (x + 5) is a factor of the function F if the remainder is 0 upon substitution.
4. So f(-5) = 0 confirms that (x + 5) is indeed a factor of the given function.