Question

1. complete each statement.

consider the function ( f(x)=3 x(x - 2)^{2}(x + 5)^{3} ). the function ( f(x) ) has
unique real zeros with a degree of
as ( x
ightarrowinfty,f(x)
ightarrow ) and
( x
ightarrow-infty,f(x)
ightarrow ). the left and right end behaviors of ( f(x) )
are
( circ ) the same
( circ ) different
because ( f(x) ) is an
( circ ) even
( circ ) odd
function.

Explanation:

Step1: Determine the degree of the polynomial

The degree of a polynomial \(f(x)=a(x - r_1)^{n_1}(x - r_2)^{n_2}\cdots(x - r_k)^{n_k}\) is \(n=n_1 + n_2+\cdots + n_k\).
For \(f(x)=3x(x - 2)^2(x + 5)^3\), the exponents are \(n_1 = 1\) (for the factor \(x\)), \(n_2=2\) (for the factor \((x - 2)^2\)) and \(n_3 = 3\) (for the factor \((x + 5)^3\)).
The degree \(n=1+2 + 3=6\).

Step2: Find the leading - term

The leading - term of the polynomial \(f(x)=3x(x - 2)^2(x + 5)^3\) is found by multiplying the leading terms of each factor.
The leading term of \(x\) is \(x\), the leading term of \((x - 2)^2=x^{2}-4x + 4\) is \(x^{2}\), and the leading term of \((x + 5)^3=x^{3}+15x^{2}+75x + 125\) is \(x^{3}\).
The leading - term of \(f(x)\) is \(3x\cdot x^{2}\cdot x^{3}=3x^{6}\).

Step3: Determine the end - behavior

For a polynomial \(y = ax^{n}\), when \(n\) is even and \(a>0\):
As \(x
ightarrow\infty\), \(y = ax^{n}
ightarrow\infty\) (since \(a = 3>0\) and \(n = 6\) (even)).
As \(x
ightarrow-\infty\), \(y=ax^{n}
ightarrow\infty\) (because \(x^{n}=(x^{2})^{3}\) and \(x^{2}>0\) for \(x
eq0\), and \(a = 3>0\)).

Step4: Find the number of unique real zeros

The zeros of the function \(f(x)=3x(x - 2)^2(x + 5)^3\) are found by setting \(f(x)=0\).
\(3x(x - 2)^2(x + 5)^3=0\) gives \(x = 0\) (with multiplicity \(1\)), \(x = 2\) (with multiplicity \(2\)), and \(x=-5\) (with multiplicity \(3\)).
The unique real zeros are \(x = 0\), \(x = 2\), and \(x=-5\), so there are \(3\) unique real zeros.

Answer:

The function \(f(x)=3x(x - 2)^2(x + 5)^3\) has \(3\) unique real zeros with a degree of \(6\). As \(x
ightarrow\infty\), \(f(x)
ightarrow\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow\infty\). The left and right - end behaviors of \(f(x)\) are the same because \(f(x)\) is an even function.