Answer:
x = - 4, 0, 3, 4
Step-by-step explanation:
Given
f(x) = - 5x(x² - 16)(x - 3) ← x² - 16 is a difference of squares, thus
f(x) = - 5x(x - 4)(x + 4)(x - 3)
To find the zeros, equate f(x) to zero, that is
- 5x(x - 4)(x + 4)(x - 3) = 0
Equate each factor to zero and solve for x
- 5x = 0 ⇒ x = 0
x - 4 = 0 ⇒ x = 4
x + 4 = 0 ⇒ x = - 4
x - 3 = 0 ⇒ x = 3
Thus the zeros are
x = - 4, x = 0, x = 3, x = 4
A multiplication equals zero if and only if at least one of the factors is zero. In this case, the factors are [tex]5x[/tex], [tex]x^2-16[/tex], [tex]x-3[/tex]. So, the equation equals zero if and only if
[tex]5x=0\iff x=0[/tex]
or
[tex]x^2-16=0\iff x^2=16 \iff x=\pm\sqrt{16}=\pm 4[/tex]
or
[tex]x-3=0\iff x=3[/tex]
