4. Find all of the zeros and multiplicities of the function 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 5 + 𝑥𝑥 4 − 19𝑥𝑥 3 + 17𝑥𝑥 2 + 48𝑥𝑥 − 60.
Begin by finding the possible rational zeros by applying the Rational Zero Theorem 𝑝𝑝
𝑞𝑞. Then graph the
function on your calculator and make predictions of rational zeros from the graph. Check your
predictions using synthetic division, and finally solve the remaining quadratic to find all the zeros.
Show all work.



To find the zeros and multiplicities of the polynomial function \( f(x) = x^5 + x^4 - 19x^3 + 17x^2 + 48x - 60 \), we can proceed with the following steps:

### Step 1: Apply the Rational Zero Theorem

The **Rational Zero Theorem** helps in determining the possible rational zeros of the polynomial. According to the theorem, the possible rational zeros are given by \( \frac{p}{q} \), where:
- \( p \) is a factor of the constant term (in this case, \( -60 \)),
- \( q \) is a factor of the leading coefficient (in this case, \( 1 \), since the coefficient of \( x^5 \) is 1).

#### Factors of the constant term \( -60 \):
The factors of \( -60 \) are:
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60
\]

#### Factors of the leading coefficient \( 1 \):
The only factor of \( 1 \) is:
\[
\pm 1
\]

Thus, the possible rational zeros are:
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60
\]

### Step 2: Graph the Function and Predict Possible Zeros

Using a graphing calculator or graphing software, you can plot the function \( f(x) = x^5 + x^4 - 19x^3 + 17x^2 + 48x - 60 \).

From the graph, the likely rational zeros are \( x = 2 \) and \( x = -3 \). 

### Step 3: Use Synthetic Division to Verify the Predictions

#### Synthetic Division with \( x = 2 \):
Let's test \( x = 2 \) using synthetic division.

The coefficients of the polynomial \( f(x) \) are: \( 1, 1, -19, 17, 48, -60 \).

Set up synthetic division:

\[
\begin{array}{r|rrrrrr}
2 & 1 & 1 & -19 & 17 & 48 & -60 \\
  &   & 2 & 6  & -26 & -18 & 60 \\
\hline
  & 1 & 3 & -13 & -9 & 30 & 0 \\
\end{array}
\]

Since the remainder is 0, \( x = 2 \) is a zero of the polynomial.

The quotient is \( x^4 + 3x^3 - 13x^2 - 9x + 30 \).

#### Synthetic Division with \( x = -3 \):
Next, let's test \( x = -3 \) using synthetic division on the quotient \( x^4 + 3x^3 - 13x^2 - 9x + 30 \).

Set up synthetic division:

\[
\begin{array}{r|rrrrr}
-3 & 1 & 3 & -13 & -9 & 30 \\
   &   & -3 & 0  & 39  & -90 \\
\hline
   & 1 & 0 & -13 & 30  & 0 \\
\end{array}
\]

Since the remainder is 0, \( x = -3 \) is a zero of the polynomial.

The quotient is \( x^3 - 13x + 10 \).

### Step 4: Solve the Remaining Polynomial \( x^3 - 13x + 10 \)

Now we need to find the zeros of the cubic polynomial \( x^3 - 13x + 10 \).

We can try the possible rational zeros again: \( \pm 1, \pm 2, \pm 5, \pm 10 \).

#### Synthetic Division with \( x = 1 \):
Let's test \( x = 1 \).

\[
\begin{array}{r|rrrr}
1 & 1 & 0 & -13 & 10 \\
  &   & 1 & 1  & -12 \\
\hline
  & 1 & 1 & -12 & -2 \\
\end{array}
\]

Since the remainder is not 0, \( x = 1 \) is not a zero.

#### Synthetic Division with \( x = -1 \):
Let's test \( x = -1 \).

\[
\begin{array}{r|rrrr}
-1 & 1 & 0 & -13 & 10 \\
   &   & -1 & 1  & -12 \\
\hline
   & 1 & -1 & -12 & -2 \\
\end{array}
\]

Since the remainder is not 0, \( x = -1 \) is not a zero.

#### Synthetic Division with \( x = 2 \):
Let's test \( x = 2 \).

\[
\begin{array}{r|rrrr}
2 & 1 & 0 & -13 & 10 \\
  &   & 2 & 4  & -18 \\
\hline
  & 1 & 2 & -9  & -8 \\
\end{array}
\]

Since the remainder is not 0, \( x = 2 \) is not a zero.

#### Synthetic Division with \( x = -2 \):
Let's test \( x = -2 \).

\[
\begin{array}{r|rrrr}
-2 & 1 & 0 & -13 & 10 \\
   &   & -2 & 4  & -18 \\
\hline
   & 1 & -2 & -9 & 2  \\
\end{array}
\]

Since the remainder is not 0, \( x = -2 \) is not a zero.

### Step 

Find all of the zeros and multiplicities of the function f(x) = x^5 + x^4 − 19x^3 + 17x^2 + 48x − 60. Begin by finding the possible rational zeros by applying the Rational Zero Theorem. Then graph the function on your calculator and make predictions of rational zeros from the graph. Check your predictions using synthetic division, and finally solve the remaining quadratic to find all the zeros.

Explore how to find all zeros and their multiplicities for the polynomial function f(x) = x^5 + x^4 − 19x^3 + 17x^2 + 48x − 60. This step-by-step solution applies the Rational Zero Theorem, synthetic division, and quadratic solving techniques. Perfect for Grades 10-12 students learning advanced algebra concepts.

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