The zeros of
y = x(x + 2)(x + 5)y=x(x+2)(x+5)
are
x=0,x=-2,x=-5x=0,x=−2,x=−5
EXPLANATION
The given function is
y = x(x + 2)(x + 5)y=x(x+2)(x+5)
To find the zeros of the above function, we just have to equate the function to zero and solve for x.
x(x + 2)(x + 5) = 0x(x+2)(x+5)=0
This implies that,
x = 0 \: \: or \: x + 2 = 0 \:or \: x + 5 = 0x=0orx+2=0orx+5=0
x = 0 \: \: or \: x = - 2\:or \: x = - 5x=0orx=−2orx=−5
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,
x < - 5x<−5
When
x = - 10x=−10
y = - 10( - 8)( - 5) \: < \: 0y=−10(−8)(−5)<0
The graph is below the x-axis.
- 5 \: < \: x \: < \: - 2−5<x<−2
When
x = - 3x=−3
y = - 3(2)( - 1) \: > 0y=−3(2)(−1)>0
The graph is above the x-axis.
- 2 \: < \: x < \: 0−2<x<0
when
x = - 1x=−1
y = -1(1)(4) \: < \: 0y=−1(1)(4)<0
The graph is below the x-axis.
Finally the interval,
x \: > \: 0x>0
when
x = 1x=1
y = 1(3)(6) \: > \: 0y=1(3)(6)>0
The graph is above the x-axis.
We can now use the above information to sketch graph as shown in the diagram above
.