Math Problem Statement
Solution
To solve this problem, we need to express the polynomial in factored form using its complex zeros. Here’s the step-by-step solution:
Step 1: Recognize the Complex Zeros
The complex zeros of are given as: These are the roots of the polynomial.
Step 2: Write in Factored Form
Since we know the roots, we can write as the product of terms of the form for each root:
Step 3: Simplify the Complex Conjugate Terms
Notice that involves a pair of complex conjugates, which we can simplify as follows: Since , we get: Expanding :
Step 4: Write the Final Factored Form
Now we can express completely in factored form:
This is the complete factorization of using its complex zeros.
Would you like more details or have any questions?
Here are five related questions to expand on this problem:
- How do you factor other polynomials with real and complex roots?
- What is the significance of complex conjugate pairs in polynomial factorization?
- How do you derive the complex roots of a cubic equation like ?
- Why is it useful to express a polynomial in its factored form?
- How does one determine if a polynomial can be factored into linear factors over complex numbers?
Tip: When dealing with polynomials that have complex roots, always look for complex conjugate pairs to simplify the factorization process.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Complex Numbers
Polynomial Factorization
Formulas
Cubic equation factorization using roots
(x - root) form for factorization
Conjugate multiplication for complex roots
Theorems
Complex Conjugate Root Theorem
Suitable Grade Level
Grades 10-12