Question

If the three real zeros of a function f(x) are located at x = r, x = q, and x = p, determine (if possible) the zeros of the function f(x -5).

Answer 1

From the question;

we are give the three zeros of the function f(x) to be located as

$x\text{ = r, x = q and x = p}$

since r, q and p are zeros of the function f(x) then

$(x\text{ - r), ( x - q) and ( x - p)}$

are factors of the function f(x)

Therefore

f(x) will be the product of the three factors

$F(x)\text{ =(x - r)(x - q)(x - p)}$

Given f(x) we can get f(x - 5) as

$\begin{gathered} f(x\text{ - 5) = (x - 5 - r)(x - 5 - q)(x - 5 - p)} \\ f(x\text{ - 5) =}(x\text{ - (5 + r))(x - (5 + q))(x - (5 + p))} \end{gathered}$

To get the zeros of the function f(x - 5) we equate it to zero

Therefore

$(x\text{ - (5 + r))(x - (5 + q))(x - (5 + p)) = 0}$

hence

$\begin{gathered} (x\text{ - (5 + r)) = 0 implies x = 5 + r} \\ (x\text{ - (5 + q)) = 0 implies x = 5 + q} \\ (x\text{ -(5 + p)) = 0 implies x = 5 + p} \end{gathered}$

Therefore, the zeros of the function f(x - 5) are

5 + r, 5 + q and 5 + p