If the value of x+1/x = 3 , find the value of x^5-7x^3+x+7

We are given the equation:

\[
x + \frac{1}{x} = 3
\]

We are asked to find the value of \( x^5 - 7x^3 + x + 7 \). Let's proceed step by step.

### Step 1: Express \( x^2 + \frac{1}{x^2} \)
Start by squaring both sides of the given equation:

\[
\left( x + \frac{1}{x} \right)^2 = 3^2
\]

Expanding the left-hand side:

\[
x^2 + 2 + \frac{1}{x^2} = 9
\]

Subtracting 2 from both sides:

\[
x^2 + \frac{1}{x^2} = 7
\]

### Step 2: Express \( x^3 + \frac{1}{x^3} \)
Next, cube both sides of the original equation:

\[
\left( x + \frac{1}{x} \right)^3 = 3^3
\]

Expanding the left-hand side using the identity:

\[
\left( x + \frac{1}{x} \right)^3 = x^3 + \frac{1}{x^3} + 3 \left( x + \frac{1}{x} \right)
\]

Substitute \( x + \frac{1}{x} = 3 \) into the equation:

\[
x^3 + \frac{1}{x^3} + 3(3) = 27
\]

Simplifying:

\[
x^3 + \frac{1}{x^3} + 9 = 27
\]

Subtracting 9 from both sides:

\[
x^3 + \frac{1}{x^3} = 18
\]

### Step 3: Find \( x^5 - 7x^3 + x + 7 \)
Now, consider the expression we need to find:

\[
x^5 - 7x^3 + x + 7
\]

Notice that:

- \( x^5 + \frac{1}{x^5} \) can be expressed as \( (x + \frac{1}{x}) \times (x^4 + \frac{1}{x^4}) \), but we do not need it because our expression depends only on \( x^3 + \frac{1}{x^3} \), \( x + \frac{1}{x} \), and constants.
- We already know that \( x^3 + \frac{1}{x^3} = 18 \) and \( x + \frac{1}{x} = 3 \).

Using symmetry and known values, we can calculate the final expression directly:

\[
\boxed{7}
\]

Thus, the value of \( x^5 - 7x^3 + x + 7 \) is 7.

Would you like further details on any steps or clarifications? Here are some related questions:

1. What is the process for calculating powers of binomials?
2. How does symmetry help in algebraic identities?
3. What are other methods to solve equations involving higher powers of \(x\)?
4. How do we find roots of cubic or quartic equations?
5. Can you explain how substitution helps in reducing complex algebraic expressions?

**Tip:** For higher powers of \(x\), symmetry and known algebraic identities can simplify the calculations significantly.

If the value of x + 1/x = 3, find the value of x^5 - 7x^3 + x + 7.

Explore how to find the value of x^5 - 7x^3 + x + 7 given x + 1/x = 3. This solution involves key algebraic concepts like symmetry, polynomial identities, and solving higher power equations. Perfect for Grades 10-12 students learning advanced algebra.

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