How many x-intercepts does the function f(x)=3(2x−1)(x+2)(8x−5)2

Question

How many x-intercepts does the function  f(x)=3(2x−1)(x+2)(8x−5)2
  have when graphed? (1 point)

GPT-4o mini · Answer

To find the number of x-intercepts of the function $f (x) = 3 (2 x - 1) (x + 2) (8 x - 5)^{2}$ , we need to determine the values of $x$ for which $f (x) = 0$ .

The function can be set equal to zero as follows:

$3 (2 x - 1) (x + 2) (8 x - 5)^{2} = 0$

Since the factor of 3 is a constant and does not affect the roots, we can focus on the polynomial part:

$(2 x - 1) (x + 2) (8 x - 5)^{2} = 0$

Now we will identify the roots of each factor:

From $2 x - 1 = 0$ : $2 x = 1 ⟹ x = \frac{1}{2}$
From $x + 2 = 0$ : $x = - 2$
From $(8 x - 5)^{2} = 0$ : $8 x - 5 = 0 ⟹ 8 x = 5 ⟹ x = \frac{5}{8}$ The factor $(8 x - 5)^{2}$ gives a double root at $x = \frac{5}{8}$ .

Now we summarize the roots:

$x = \frac{1}{2}$ (single root)
$x = - 2$ (single root)
$x = \frac{5}{8}$ (double root)

X-intercepts correspond to the unique values of $x$ where the function is equal to zero. The roots we have found are:

$x = \frac{1}{2}$
$x = - 2$
$x = \frac{5}{8}$

Counting the unique roots, we have three distinct x-intercepts:

$\frac{1}{2}$
$- 2$
$\frac{5}{8}$

Thus, the function $f (x)$ has 3 x-intercepts.