Let's analyze the polynomial function given: [tex]\( f(x) = 4x^7 + 40x^6 + 100x^5 \)[/tex].
### Step 1: Factor the Polynomial
First, we can factor out the greatest common divisor:
[tex]\[ f(x) = 4x^5(x^2 + 10x + 25) \][/tex]
### Step 2: Further Factor the Expression
Notice that [tex]\( x^2 + 10x + 25 \)[/tex] is a perfect square trinomial:
[tex]\[ x^2 + 10x + 25 = (x + 5)^2 \][/tex]
So, the given polynomial can be factored as:
[tex]\[ f(x) = 4x^5(x + 5)^2 \][/tex]
### Step 3: Identify the Roots
The roots of the polynomial are where [tex]\( f(x) = 0 \)[/tex]. Setting the factored form to zero:
[tex]\[ 4x^5(x + 5)^2 = 0 \][/tex]
We get the roots:
[tex]\[ x = 0 \quad \text{and} \quad x = -5 \][/tex]
### Step 4: Determine the Multiplicity of Each Root
The multiplicity of a root is the power of the factor corresponding to that root.
- For [tex]\( x = 0 \)[/tex]: The factor is [tex]\( x^5 \)[/tex], so the multiplicity is 5 (odd).
- For [tex]\( x = -5 \)[/tex]: The factor is [tex]\( (x + 5)^2 \)[/tex], so the multiplicity is 2 (even).
### Step 5: Analyze the Graph Behavior at Each Root
- A root with an odd multiplicity indicates that the graph crosses the x-axis at that point.
- A root with an even multiplicity indicates that the graph touches the x-axis at that point but does not cross it.
### Conclusion:
- The graph crosses the x-axis at [tex]\( x = 0 \)[/tex] (odd multiplicity of 5).
- The graph touches the x-axis at [tex]\( x = -5 \)[/tex] (even multiplicity of 2).
Therefore, the correct statement is:
- The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x=0 \)[/tex] and touches the [tex]\( x \)[/tex]-axis at [tex]\( x=-5 \)[/tex].
Hence, the correct answer is:
'The graph crosses the [tex]$x$[/tex]-axis at [tex]$x=0$[/tex] and touches the [tex]$x$[/tex]-axis at [tex]$x=-5$[/tex].'
