We are given the following system of equations:

$a + b + c = 4$ $a^2 + b^2 + c^2 = 10$ $a^3 + b^3 + c^3 = 22$

Our goal is to determine the value of:

$a^4 + b^4 + c^4 = ?$

This problem can be approached systematically using Newton’s sums, which relate the symmetric sums of roots with the powers of roots. Let's go step-by-step to solve for $a^4 + b^4 + c^4$.

Step-by-Step Solution using Newton's Sums

We define the symmetric sums:

• $P_1 = a + b + c = 4$,
• $P_2 = a^2 + b^2 + c^2 = 10$,
• $P_3 = a^3 + b^3 + c^3 = 22$,
• We want to find $P_4 = a^4 + b^4 + c^4$.

Newton's sums formula for higher-order terms:

$P_n = (a + b + c) P_{n-1} - \sum_{k=1}^{n-2} P_k \cdot S_{n-k} + n \cdot S_n$

Since we are dealing with three terms $a, b, c$, the coefficients $S_k$ for $k > 1$ are generally related to sums of products like $ab + bc + ca$.

Step 1: Use identity for squares.

We have the identity:

$a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca)$

Plugging in known values:

$10 = 4^2 - 2(ab + bc + ca)$

$10 = 16 - 2(ab + bc + ca)$

$2(ab + bc + ca) = 6 \quad \Rightarrow \quad ab + bc + ca = 3$

Step 2: Express higher powers.

Now, using the Newton's sum formula for $P_3$, we get:

$P_3 = (a + b + c) P_2 - (ab + bc + ca) P_1$

Plugging in values:

$22 = 4 \cdot 10 - 3 \cdot 4$

$22 = 40 - 12$

This is consistent. Now let’s use the same formula to find $P_4$.

Step 3: Find $P_4$.

Using the Newton's sum formula for $P_4$:

$P_4 = (a + b + c) P_3 - (ab + bc + ca) P_2$

Substitute the known values:

$P_4 = 4 \cdot 22 - 3 \cdot 10$

$P_4 = 88 - 30 = 58$

Final Answer

$a^4 + b^4 + c^4 = 58$

Do you need further explanation, or would you like to explore another method?
Here are some related questions for practice:

1. How do you derive Newton's sums for polynomial roots in general?
2. What are the identities for $a^5 + b^5 + c^5$ using the same approach?
3. How is the symmetric sum $ab + bc + ca$ derived from roots of a cubic equation?
4. What are the limits of Newton’s sums? Can they be applied to non-polynomial roots?
5. How would you solve this system if you knew the explicit roots $a, b,$ and $c$?

Tip: Newton’s sums are very useful for symmetric polynomial problems. Mastering them helps solve high-degree equations efficiently.