How many x -intercepts would the function shown below have? f(x)=2x(x+7)(x^2-7)(4x+5)^2(x+7) Answer Attempt 1 out of 2

To find the x-intercepts of the function, we need to set the function equal to zero and solve for x.<br /><br />$f(x) = 2x(x+7)(x^{2}-7)(4x+5)^{2}(x+7)$<br /><br />Setting the function equal to zero:<br /><br />$2x(x+7)(x^{2}-7)(4x+5)^{2}(x+7) = 0$<br /><br />Now, we can solve for x by setting each factor equal to zero:<br /><br />1. $2x = 0$<br /> Solving for x, we get $x = 0$.<br /><br />2. $(x+7) = 0$<br /> Solving for x, we get $x = -7$.<br /><br />3. $(x^{2}-7) = 0$<br /> Solving for x, we get $x = \sqrt{7}$ and $x = -\sqrt{7}$.<br /><br />4. $(4x+5)^{2} = 0$<br /> Solving for x, we get $x = -\frac{5}{4}$.<br /><br />5. $(x+7) = 0$<br /> Solving for x, we get $x = -7$ (repeated).<br /><br />Therefore, the function has 5 x-intercepts: $x = 0$, $x = -7$, $x = \sqrt{7}$, $x = -\sqrt{7}$, and $x = -\frac{5}{4}$.<br /><br />Answer: The function has 5 x-intercepts.