One rational zero of the given function f(x)=7x^3+52x^2+22x+7

To find the rational zero of the given cubic function \( f(x) = 7x^3 + 52x^2 + 22x + 7 \), we can apply the **Rational Root Theorem**. This theorem suggests that any rational root, in the form \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient.

### Step 1: Apply the Rational Root Theorem
- The constant term of the polynomial is \( 7 \), so \( p \) (possible factors of the constant) are: \( \pm 1, \pm 7 \).
- The leading coefficient is also \( 7 \), so \( q \) (possible factors of the leading coefficient) are: \( \pm 1, \pm 7 \).

Thus, the possible rational roots are the combinations of these factors:  
\[ \pm 1, \pm \frac{1}{7}, \pm 7 \]

### Step 2: Test possible rational roots
We test each possible rational root by substituting them into the polynomial:

1. **Testing \( x = 1 \)**:
   \[
   f(1) = 7(1)^3 + 52(1)^2 + 22(1) + 7 = 7 + 52 + 22 + 7 = 88
   \]
   Since \( f(1) \neq 0 \), \( x = 1 \) is not a root.

2. **Testing \( x = -1 \)**:
   \[
   f(-1) = 7(-1)^3 + 52(-1)^2 + 22(-1) + 7 = -7 + 52 - 22 + 7 = 30
   \]
   Since \( f(-1) \neq 0 \), \( x = -1 \) is not a root.

3. **Testing \( x = \frac{1}{7} \)**:
   \[
   f\left(\frac{1}{7}\right) = 7\left(\frac{1}{7}\right)^3 + 52\left(\frac{1}{7}\right)^2 + 22\left(\frac{1}{7}\right) + 7
   \]
   This involves fractions and is not likely to be the root, so we move on to simpler integer candidates first.

4. **Testing \( x = -7 \)**:
   \[
   f(-7) = 7(-7)^3 + 52(-7)^2 + 22(-7) + 7 = 7(-343) + 52(49) + 22(-7) + 7
   \]
   \[
   = -2401 + 2548 - 154 + 7 = 0
   \]
   Since \( f(-7) = 0 \), \( x = -7 \) is a root.

### Conclusion:
The rational zero of the function \( f(x) = 7x^3 + 52x^2 + 22x + 7 \) is \( x = -7 \).

Would you like more details on the steps, or have any questions?

### Related Questions:
1. How can we fully factor the polynomial after finding a rational zero?
2. What are the remaining roots of \( f(x) \) after finding the rational zero?
3. Can you show the synthetic division process for \( f(x) \) divided by \( x + 7 \)?
4. How do we verify if the Rational Root Theorem covers all possibilities?
5. What are the applications of finding rational zeros in real-life problems?

### Tip:
When using the Rational Root Theorem, always start testing with simple integer factors before moving on to fractions.

Discover how to apply the Rational Root Theorem to find a rational zero of the cubic function f(x) = 7x^3 + 52x^2 + 22x + 7. This step-by-step solution explains key algebraic concepts and methods, ideal for students in Grades 10-12.

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