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 f(x)f(x) has real coefficients and is of degree 5. You are given three zeros: 11, ii, and 3i-3i.

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 ii is i-i.
  • The conjugate of 3i-3i is 3i3i.

Thus, the remaining zeros are:

iand3i- i \quad \text{and} \quad 3i

So, the complete list of zeros of f(x)f(x) is: 1,i,i,3i,3i1, i, -i, -3i, 3i

Would you like further details or have any questions?

Here are 5 related questions:

  1. What is the conjugate of a complex number, and how is it used in polynomial zeros?
  2. How do real coefficients affect the zeros of a polynomial?
  3. Can a polynomial with real coefficients have an odd number of complex zeros?
  4. What is the general form of a degree 5 polynomial given these zeros?
  5. 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