Math Problem Statement
3, 1, -sqrt2
Write a polynomial function f of least degree, that has rational coefficients, a leading coefficient of 1, and the given zeros
Solution
To create a polynomial function with the given zeros , with rational coefficients and a leading coefficient of 1, follow these steps:
Step 1: Express the zeros as factors.
For each zero, there is a corresponding factor of the form :
- For , the factor is .
- For , the factor is .
- For , the factor is .
Step 2: Account for the irrational root.
Because the polynomial must have rational coefficients, the conjugate of the irrational root (which is ) must also be a root. Thus, you also have the factor .
Step 3: Write the polynomial as a product of all factors.
The polynomial is:
Step 4: Simplify the product involving the irrational roots.
Notice that: This is the difference of squares.
Step 5: Multiply all the factors.
Now, the polynomial becomes:
Step 6: Expand the polynomial.
First, multiply :
Now multiply this by :
Step 7: Perform the expansion.
Step 8: Combine like terms.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Functions
Rational Coefficients
Conjugate Roots
Formulas
Difference of squares formula: (x + sqrt{a})(x - sqrt{a}) = x^2 - a
Polynomial expansion
Theorems
Conjugate Root Theorem
Fundamental Theorem of Algebra
Suitable Grade Level
Grades 10-12