Math Problem Statement
Information is given about a polynomial f left parenthesis x right parenthesis whose coefficients are real numbers. Find the remaining zeros of f. Degree 5; zeros: 1 comma i comma negative 3 i Question content area bottom Part 1 The remaining zero(s) of f is(are)
(Use a comma to separate answers as needed.)
Solution
The polynomial has real coefficients and is of degree 5. You are given three zeros: , , and .
Since the coefficients are real, the complex zeros must come in conjugate pairs. This means that for each complex zero, its conjugate must also be a zero of the polynomial.
- The conjugate of is .
- The conjugate of is .
Thus, the remaining zeros are:
So, the complete list of zeros of is:
Would you like further details or have any questions?
Here are 5 related questions:
- What is the conjugate of a complex number, and how is it used in polynomial zeros?
- How do real coefficients affect the zeros of a polynomial?
- Can a polynomial with real coefficients have an odd number of complex zeros?
- What is the general form of a degree 5 polynomial given these zeros?
- How can you factor a polynomial using its zeros?
Tip: When working with polynomials with real coefficients, always remember that complex zeros occur in conjugate pairs.
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Math Problem Analysis
Mathematical Concepts
Polynomials
Complex Numbers
Conjugates
Zeros of Polynomials
Formulas
Complex conjugate pairs: If a + bi is a zero, its conjugate a - bi is also a zero for polynomials with real coefficients.
Theorems
Conjugate Root Theorem: If a polynomial has real coefficients, any complex zeros must occur in conjugate pairs.
Suitable Grade Level
Grades 10-12
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