Math Problem Statement
Solution
To form a polynomial with the given properties:
Given Information
- Zeros:
- with multiplicity
- with multiplicity
- Degree:
- Leading coefficient:
Solution
-
Write each factor for the given zeros and their multiplicities:
- For zero with multiplicity : the factor is .
- For zero with multiplicity : the factor is .
-
Combine the factors:
-
Expand the expression: First, expand : Now multiply by :
-
**Distribute across each term in :
-
Combine like terms:
Final Answer
The polynomial is:
This polynomial has integer coefficients, a leading coefficient of , and the correct zeros with specified multiplicities.
Would you like a deeper breakdown of any step?
Here are five related questions to deepen your understanding:
- What is the impact of a zero's multiplicity on the shape of a polynomial's graph?
- How would the polynomial change if the zero had multiplicity instead of ?
- How can you confirm the degree of the polynomial after expansion?
- How does the leading coefficient affect the end behavior of the polynomial?
- What method can you use to factor polynomials when only given some of the roots?
Tip: When expanding polynomials, keep track of each term carefully to avoid errors in sign or coefficient.
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Math Problem Analysis
Mathematical Concepts
Polynomials
Roots and Zeros
Multiplicities
Formulas
f(x) = (x - a)^m * (x - b)^n
Theorems
Fundamental Theorem of Algebra
Suitable Grade Level
Grades 10-12