Find all the real zeros of the polynomial. Separate answers by commas. Enter exact answers using integers and fractions, not decimal approximations. f ( x ) = x^4 + 11x^3 + 23x^2 − 95x − 300

To find all the real zeros of the polynomial \( f(x) = x^4 + 11x^3 + 23x^2 - 95x - 300 \), we will follow these steps:

### 1. Use Rational Root Theorem:
The Rational Root Theorem suggests that any rational root of a polynomial can be expressed as \( \frac{p}{q} \), where \( p \) is a factor of the constant term (-300) and \( q \) is a factor of the leading coefficient (1).

The factors of -300 are: 
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 25, \pm 30, \pm 50, \pm 60, \pm 75, \pm 100, \pm 150, \pm 300
\]
The factors of 1 are: \( \pm 1 \).

Thus, the possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 25, \pm 30, \pm 50, \pm 60, \pm 75, \pm 100, \pm 150, \pm 300 \).

### 2. Test potential rational roots:
We will start by substituting possible values of \( x \) from the list of rational roots into the polynomial \( f(x) \) to find any zeros.

Let's try \( x = -5 \):

\[
f(-5) = (-5)^4 + 11(-5)^3 + 23(-5)^2 - 95(-5) - 300
\]
\[
= 625 - 1375 + 575 + 475 - 300 = 0
\]

Thus, \( x = -5 \) is a root.

### 3. Polynomial Division:
Since \( x = -5 \) is a root, we can divide the polynomial \( f(x) \) by \( (x + 5) \) using synthetic or long division.

Performing synthetic division:

\[
\text{Divide } (x^4 + 11x^3 + 23x^2 - 95x - 300) \text{ by } (x + 5)
\]

\[
\begin{array}{r|rrrrr}
  -5 & 1 & 11 & 23 & -95 & -300 \\
     &   & -5 & -30 & 35  & 300 \\
\hline
     & 1 & 6  & -7  & -60 & 0 \\
\end{array}
\]

The quotient is \( x^3 + 6x^2 - 7x - 60 \).

### 4. Factor the quotient \( x^3 + 6x^2 - 7x - 60 \):
We will now apply the Rational Root Theorem again to \( x^3 + 6x^2 - 7x - 60 \).

Possible rational roots for this quotient are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60 \).

Let's try \( x = -3 \):

\[
f(-3) = (-3)^3 + 6(-3)^2 - 7(-3) - 60
\]
\[
= -27 + 54 + 21 - 60 = -12
\]

This is not a root. Now, let's try \( x = 4 \):

\[
f(4) = (4)^3 + 6(4)^2 - 7(4) - 60
\]
\[
= 64 + 96 - 28 - 60 = 72
\]

This is also not a root. Let's test \( x = -4 \):

\[
f(-4) = (-4)^3 + 6(-4)^2 - 7(-4) - 60
\]
\[
= -64 + 96 + 28 - 60 = 0
\]

Thus, \( x = -4 \) is a root.

### 5. Perform Synthetic Division again:
Now divide \( x^3 + 6x^2 - 7x - 60 \) by \( (x + 4) \):

\[
\begin{array}{r|rrrr}
  -4 & 1 & 6 & -7 & -60 \\
     &   & -4 & -8 & 60 \\
\hline
     & 1 & 2 & -15 & 0 \\
\end{array}
\]

The quotient is \( x^2 + 2x - 15 \).

### 6. Factor \( x^2 + 2x - 15 \):
Now, factor \( x^2 + 2x - 15 \):

\[
x^2 + 2x - 15 = (x - 3)(x + 5)
\]

### 7. Find the remaining roots:
The factors of \( f(x) \) are:
\[
f(x) = (x + 5)(x + 4)(x - 3)(x + 5)
\]

The real zeros are:
\[
x = -5, -4, 3
\]

Since \( (x + 5) \) appears twice, \( x = -5 \) is a repeated root.

### Final Answer:
The real zeros of \( f(x) \) are \( -5, -4, 3 \).

Would you like more details or explanations on any of the steps? 

Here are 5 related questions to expand this topic:
1. What is the Rational Root Theorem and how is it applied?
2. How does synthetic division work with polynomials?
3. Why do we perform polynomial division to find other factors?
4. What are repeated roots, and how do they affect the graph of a polynomial?
5. How can we verify the real zeros of a polynomial?

**Tip:** Always check the remainder when performing synthetic division. If it’s zero, the divisor is indeed a root!

Find all the real zeros of the polynomial. Separate answers by commas. Enter exact answers using integers and fractions, not decimal approximations. f(x) = x^4 + 11x^3 + 23x^2 − 95x − 300

Discover how to find all the real zeros of the polynomial f(x) = x^4 + 11x^3 + 23x^2 − 95x − 300 using the Rational Root Theorem, synthetic division, and factoring. This step-by-step guide is suitable for high school students (Grades 10-12) and covers essential algebraic techniques.

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