Question

which statement describes the graph of f(x) = 4x⁷ + 40x⁶ + 100x⁵?
○ the graph crosses the x-axis at x = 0 and touches the x-axis at x = 5.
○ the graph touches the x-axis at x = 0 and crosses the x-axis at x = 5.
○ the graph crosses the x-axis at x = 0 and touches the x-axis at x = −5.
○ the graph touches the x-axis at x = 0 and crosses the x-axis at x = −5.

Explanation:

Step1: Factor the function

First, factor out the greatest common factor from \( f(x) = 4x^{7}+40x^{6}+100x^{5} \). The GCF is \( 4x^{5} \), so we get:
\( f(x)=4x^{5}(x^{2} + 10x + 25) \)

Step2: Factor the quadratic

Next, factor the quadratic \( x^{2}+10x + 25 \). We know that \( x^{2}+10x + 25=(x + 5)^{2} \) (since it's a perfect square trinomial, \( (a + b)^{2}=a^{2}+2ab + b^{2} \) with \( a=x \) and \( b = 5 \)). So the function becomes:
\( f(x)=4x^{5}(x + 5)^{2} \)

Step3: Analyze the roots

For a polynomial function, the roots are the values of \( x \) where \( f(x)=0 \).

• For the factor \( x^{5} \), the root is \( x = 0 \) with multiplicity \( 5 \) (which is odd). When the multiplicity of a root is odd, the graph crosses the \( x \)-axis at that root.
• For the factor \( (x + 5)^{2} \), the root is \( x=- 5 \) with multiplicity \( 2 \) (which is even). When the multiplicity of a root is even, the graph touches the \( x \)-axis at that root (and turns around).

So the graph crosses the \( x \)-axis at \( x = 0 \) (because multiplicity 5 is odd) and touches the \( x \)-axis at \( x=-5 \) (because multiplicity 2 is even).

Answer:

The graph crosses the x - axis at \( x = 0 \) and touches the x - axis at \( x=-5 \) (the third option: The graph crosses the x - axis at \( x = 0 \) and touches the x - axis at \( x=-5 \))