Math Problem Statement
A polynomial function f(x) with real coefficients has the given degree, zeros, and solution point.
Degree
Zeros
Solution Point
4
−1, 5, i
f(1) = 48
(a) Write the function in completely factored form.
f(x) =
Solution
To find the polynomial function with the given information, we proceed as follows:
Given:
- Degree: 4
- Zeros: , , and
- Solution point:
Since the polynomial has real coefficients, the complex conjugate must also be a zero. The zeros are , , , and .
Step 1: Form the factors
The zeros give us the factors of the polynomial:
Step 2: Write the function in factored form
The polynomial can be written as: where is a constant to be determined.
Step 3: Simplify the complex conjugate factors
Notice that: Thus, the polynomial becomes:
Step 4: Determine the value of
Use the given solution point :
Step 5: Write the final function
The polynomial in completely factored form is:
Would you like details on how to expand this polynomial, or have any other questions?
Related Questions:
- How would you expand into standard form?
- What happens to the zeros if the polynomial's coefficients are not real?
- How do complex conjugate pairs affect the roots of a polynomial?
- What would be the steps to find the derivative of ?
- How can you verify that satisfies ?
Tip:
When given zeros of a polynomial with real coefficients, always include the conjugate pair for any complex zeros.
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Math Problem Analysis
Mathematical Concepts
Polynomial Functions
Complex Numbers
Real Coefficients
Factoring
Formulas
f(x) = a(x + 1)(x - 5)(x - i)(x + i)
(x - i)(x + i) = x^2 + 1
Theorems
Fundamental Theorem of Algebra
Complex Conjugate Root Theorem
Suitable Grade Level
Grades 10-12
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