https://file.aiquickdraw.com/mathbot/file/1729274542969ktt04ckq.jpg

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.

Given the system of equations: a + b + c = 4, a^2 + b^2 + c^2 = 10, a^3 + b^3 + c^3 = 22, find a^4 + b^4 + c^4.

Learn how to find a^4 + b^4 + c^4 using Newton's sums given a + b + c = 4, a^2 + b^2 + c^2 = 10, and a^3 + b^3 + c^3 = 22. This detailed solution covers key algebraic concepts and step-by-step calculations. Ideal for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation