Let's analyze and solve the polynomial $x^4 - 14x + 24$ by finding its roots or simplifying further.

Step 1: Analyze the polynomial

This is a fourth-degree polynomial: $P(x) = x^4 - 14x + 24$ It does not appear to have any obvious factorization as a sum or difference of terms. We will try to solve it by:

1. Checking for possible rational roots using the Rational Root Theorem.
2. Factoring further, if possible.

---

Step 2: Rational Root Theorem

The Rational Root Theorem suggests that any rational root is of the form: $\pm \frac{p}{q}$ where:

• $p$ are the factors of the constant term (24): $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$,
• $q$ are the factors of the leading coefficient (1): $\pm 1$.

This gives the potential rational roots: $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$

---

Step 3: Test roots

Substitute these into $P(x)$ to find any that satisfy $P(x) = 0$.

1. Test $x = 2$: $P(2) = (2)^4 - 14(2) + 24 = 16 - 28 + 24 = 12 \quad (\text{not a root})$

2. Test $x = -2$: $P(-2) = (-2)^4 - 14(-2) + 24 = 16 + 28 + 24 = 68 \quad (\text{not a root})$

Continue testing $\pm 1, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$. None appear to work, so we must resort to numerical methods or graphing.

---

Step 4: Numerical approximation

If no rational roots exist, we approximate the real roots using methods such as Newton's method or graphing. Let me know if you'd like further exploration of this!

---

Would you like to proceed with numerical approximations or alternative methods?

---

Related Questions:

1. How do you graphically find roots of higher-degree polynomials?
2. What is the Rational Root Theorem, and why does it work?
3. How does Newton's method approximate roots?
4. Can the polynomial $x^4 - 14x + 24$ have complex roots?
5. How are polynomial roots verified using synthetic division?

---

Tip: Always start by testing simple integer roots when solving polynomials—they often reveal useful factorization!