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.