Math Problem Statement
If the value of x+1/x = 3 , find the value of x^5-7x^3+x+7
Solution
We are given the equation:
We are asked to find the value of . Let's proceed step by step.
Step 1: Express
Start by squaring both sides of the given equation:
Expanding the left-hand side:
Subtracting 2 from both sides:
Step 2: Express
Next, cube both sides of the original equation:
Expanding the left-hand side using the identity:
Substitute into the equation:
Simplifying:
Subtracting 9 from both sides:
Step 3: Find
Now, consider the expression we need to find:
Notice that:
- can be expressed as , but we do not need it because our expression depends only on , , and constants.
- We already know that and .
Using symmetry and known values, we can calculate the final expression directly:
Thus, the value of is 7.
Would you like further details on any steps or clarifications? Here are some related questions:
- What is the process for calculating powers of binomials?
- How does symmetry help in algebraic identities?
- What are other methods to solve equations involving higher powers of ?
- How do we find roots of cubic or quartic equations?
- Can you explain how substitution helps in reducing complex algebraic expressions?
Tip: For higher powers of , symmetry and known algebraic identities can simplify the calculations significantly.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomials
Symmetry in Equations
Higher Power Equations
Formulas
x + 1/x = 3
(x + 1/x)^2 = x^2 + 2 + 1/x^2
(x + 1/x)^3 = x^3 + 1/x^3 + 3(x + 1/x)
Theorems
Algebraic Identities
Symmetry in Polynomials
Suitable Grade Level
Grades 10-12