crosses x-axis at -2 and 3 touches the x-axis at 5, has a y-intercept of -5 and is below the x-axis between -2 and 3

Question

1. crosses x-axis at -2 and 3
2. touches the x-axis at 5,
3. has a y-intercept of -5
4. and is below the x-axis between -2 and 3

Answer 1

If a root crosses the x axis it has a multiplicity of 1. If it touches the x Axis it has a multiplicity of 2 so you can write the equation in factored form as y= a(x+2)(x-3)(x-5)^2

If the y intercept is at -5 you have a point on the graph of (0.-5) which you can use to,solve for a

-5=A( 0+2)(0-3)(0-5)^2

-5= A(2)(-3)(25)

-5 =-150a

a=1/30

so equation is y= (1/30)(x+2)(x-3)(x-5)^2