


Find all rational zeros of the given polynomial function.

​f(x)equals=x^3+7x^2+7x-15 

To find the rational zeros of the polynomial function \( f(x) = x^3 + 7x^2 + 7x - 15 \), we can apply the **Rational Root Theorem**. This theorem suggests that any rational root of the polynomial is of the form \( \frac{p}{q} \), where:

- \( p \) is a factor of the constant term (-15), and
- \( q \) is a factor of the leading coefficient (1, since the leading term is \( x^3 \)).

### Step 1: Identify possible rational roots
The constant term is -15, and the factors of -15 are:
\[
p = \pm 1, \pm 3, \pm 5, \pm 15
\]

The leading coefficient is 1, and the factors of 1 are:
\[
q = \pm 1
\]

So, the possible rational roots are \( \pm 1, \pm 3, \pm 5, \pm 15 \).

### Step 2: Test the possible rational roots using synthetic division or direct substitution

We test each of the possible rational roots by substituting them into the polynomial to check if \( f(x) = 0 \).

- \( f(1) = 1^3 + 7(1)^2 + 7(1) - 15 = 1 + 7 + 7 - 15 = 0 \)

Since \( f(1) = 0 \), \( x = 1 \) is a root.

### Step 3: Perform synthetic division to factor the polynomial

Now that we know \( x = 1 \) is a root, we can divide \( f(x) \) by \( x - 1 \) using synthetic division.

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

The quotient is \( x^2 + 8x + 15 \).

### Step 4: Factor the quotient

Now, we factor the quadratic \( x^2 + 8x + 15 \):

\[
x^2 + 8x + 15 = (x + 3)(x + 5)
\]

### Step 5: Find all roots

So, the complete factorization of \( f(x) \) is:

\[
f(x) = (x - 1)(x + 3)(x + 5)
\]

The rational zeros of \( f(x) \) are:
\[
x = 1, -3, -5
\]

### Final Answer:
The rational zeros of \( f(x) = x^3 + 7x^2 + 7x - 15 \) are \( x = 1, -3, -5 \).

Do you want more details on any step or have any questions?

### Related Questions:
1. How can synthetic division be used to check other possible rational roots?
2. What is the importance of the Rational Root Theorem in solving polynomial equations?
3. How would the process change if the leading coefficient was not 1?
4. Can you explain the factorization process of quadratics in more detail?
5. How do we determine irrational roots of a polynomial?

### Tip:
Always start by applying the Rational Root Theorem to reduce the complexity of polynomial equations before moving on to more advanced techniques like factoring or using the quadratic formula.

Find all rational zeros of the given polynomial function.
​f(x) = x^3 + 7x^2 + 7x - 15

Explore the process of finding rational zeros of the polynomial function f(x) = x^3 + 7x^2 + 7x - 15 using the Rational Root Theorem. This step-by-step solution involves synthetic division and factorization techniques, ideal for high school students in Grades 9-12.

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