Given F(x)=2x(x-3)^3(x+5)^2

Question

Given F(x)=2x(x-3)^3(x+5)^2

 

A.) Find the leading term

B.) Describe the end-behavior of the function F

C.) FInd all the zeros of F, and state their multiplicity

D.) State whether the graph crosses, or only touches the x-axis at each zero 

Answer 1

A)

 

The leading term is the term with the highest exponent of x.  If we expand F(x).

 

 

F(x) = 2x(x - 3)(x - 3)(x - 3)(x + 5)(x + 5)

 

F(x) = 2x(x² - 6x + 9)(x - 3)(x² + 10x + 25)

 

F(x) = (2x³ - 12x² + 18x)(x³ + 10x² + 25x - 3x² - 30x - 75)

 

F(x) = (2x³ - 12x² + 18x)(x³ + 7x² - 5x - 75)

 

 

Multiply the first terms of the polynomials to get 2x⁶.  This is your leading term.

 

 

B)

 

The leading term determines the shape of the graph.  Since the leading term is positive, and the degree of F(x) is even, the graph is even and starts to decrease from the left endpoint.  Then it increases at the right endpoint.

 

 

C)

 

To find the zeros, we set f(x) equal to zero.   The multiplicity is the number of times the zeros repeat. From the equation, we can see that

 

x = 0 , multiplicity of 1.

x = 3 , multiplicity of 3.

x = - 5 , multiplicity of 2.

 

Note that when we add the multiplicities, we have 6 multiplicities.  This number is the same as the degree of F(x).

 

 

D)

 

If a zero has an even multiplicity, then F(x) touches the x-axis at the zero.

If a zero has an odd multiplicity, then F(x) crosses the x-axis at the zero.

 

F(x) crosses the x-axis at x=0  and  x=3.

F(x) touches the x-axis at x=-5.