ANSWER
The zeros of
[tex]y = x(x + 2)(x + 5)[/tex]
are
[tex]x=0,x=-2,x=-5[/tex]
EXPLANATION
The given function is
[tex]y = x(x + 2)(x + 5)[/tex]
To find the zeros of the above function, we just have to equate the function to zero and solve for x.
[tex]x(x + 2)(x + 5) = 0[/tex]
This implies that,
[tex]x = 0 \: \: or \: x + 2 = 0 \:or \: x + 5 = 0[/tex]
[tex]x = 0 \: \: or \: x = - 2\:or \: x = - 5[/tex]
To graph the above function, we need to consider the multiplicity.
We can see that the multiplicity of the roots are odd. This means that, the graph crosses the x-axis at each x-intercept.
We also need to consider the position of the graph on the following intervals,
[tex]x < - 5[/tex]
When
[tex]x = - 10[/tex]
[tex]y = - 10( - 8)( - 5) \: < \: 0[/tex]
The graph is below the x-axis.
[tex] - 5 \: < \: x \: < \: - 2[/tex]
When
[tex]x = - 3[/tex]
[tex]y = - 3(2)( - 1) \: > 0[/tex]
The graph is above the x-axis.
[tex] - 2 \: < \: x < \: 0[/tex]
when
[tex]x = - 1[/tex]
[tex]y = -1(1)(4) \: < \: 0[/tex]
The graph is below the x-axis.
Finally the interval,
[tex]x \: > \: 0[/tex]
when
[tex]x = 1[/tex]
[tex]y = 1(3)(6) \: > \: 0[/tex]
The graph is above the x-axis.
We can now use the above information to sketch graph as shown in the diagram above.
