Math Problem Statement
α and β are the zeros of the quadratic polynomial x² - 5x + 6, find the value of α⁴ + β² + α² + β⁴.
Solution
The given quadratic polynomial is:
Step 1: Identify sum and product of roots
From the standard form , we compare:
- Sum of roots:
- Product of roots:
Step 2: Compute
Using the identity:
First, calculate using:
Next, calculate using:
Now, substitute these values:
Thus, the required value is:
Would you like further clarification or step-by-step breakdowns? Here are some related questions to explore:
- How do we derive the identities used in solving this problem?
- Can we generalize this for any quadratic equation?
- What happens if the quadratic polynomial has complex roots?
- How do we extend this method to higher-degree polynomials?
- Can we solve this problem using a different approach?
Tip: Always verify your algebraic identities before applying them in calculations!
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Roots of Polynomials
Power of Terms
Formulas
Sum of roots: α + β = -b/a
Product of roots: αβ = c/a
Identity for powers: α⁴ + β⁴ = (α² + β²)² - 2α²β²
Theorems
Vieta's Formulas
Quadratic Identity
Suitable Grade Level
Grades 9-11