Question

given (f(x)=3(x + 1)^{4}(x - 5)(3x - 5)^{6}(x + 2)). list the zeros (ex. 4, 6, -2): degree: end behavior: as (x\to-infty), (f(x)\to) select an answer (>) as (x\to+infty), (f(x)\to) select an answer (>)

Explanation:

Step1: Find the zeros

Set each factor equal to zero.
For \(x + 1=0\), we get \(x=-1\); for \(x - 5=0\), we get \(x = 5\); for \(3x-5 = 0\), we solve \(3x=5\) and get \(x=\frac{5}{3}\); for \(x + 2=0\), we get \(x=-2\).
The zeros are \(x=-2,-1,\frac{5}{3},5\).

Step2: Determine the degree

The degree of a polynomial is the sum of the exponents of its factors. Here the exponents are all 1 (since the factors are linear), and we have 4 factors, so the degree is \(1 + 1+1 + 1=4\).

Step3: Analyze end - behavior

The leading coefficient of the polynomial \(f(x)=3(x + 1)(x - 5)(3x - 5)(x + 2)\) is the product of the leading coefficients of each factor, \(3\times1\times3\times1 = 9>0\). Since the degree \(n = 4\) (even), as \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\); as \(x
ightarrow+\infty\), \(f(x)
ightarrow+\infty\).

Answer:

Zeros: \(x=-2,-1,\frac{5}{3},5\)
Degree: 4
As \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\)
As \(x
ightarrow+\infty\), \(f(x)
ightarrow+\infty\)