Question

Fins the zeros, multiplicity, and effect on the graph of the function in

f(x) = -x (3x-2)^2(x+9)^5
and
f(x) = x^3+10x^2+25x

Answer 1

See explanation

Explanation:

Zeroe of the function is such velue of x at which f(x)=0.

1. Consider the function
`f(x)=-x(3x-2)^2 (x+9)^5.`

Zeros are:

`-x(3x-2)^2(x+9)^5=0\\ \\x=0\text{ or }x=(2)/(3)\text{ or }x=-9.`

Zero
`x=0` has multiplicity of 1, zero
`x=(2)/(3)` has multiplicity of 2, zero
`x=-9` has multiplicity of 5.

At
`x=0` or
`x=-9` the graph of the function crosses the x-axis, at
`x=(2)/(3)` the graph of the function touches the x-axis.

2. Consider the function
`f(x)=x^3+10x^2+25x=x(x^2+10x+25)=x(x+5)^2.`

Zeros are:

`x(x+5)^2=0\\ \\x=0\text{ or }x=-5.`

Zero
`x=0` has multiplicity of 1, zero
`x=-5` has multiplicity of 2.

At
`x=0` the graph of the function crosses the x-axis, at
`x=-5` the graph of the function touches the x-axis.