Math Problem Statement
Use the given zero to find the remaining zeros of the polynomial function. (Enter your answers as a comma-separated list.) P(x) = x3 + 9x2 + x + 9; −i x =
Solution
We are given the polynomial function:
and one of its zeros, .
Step 1: Complex Conjugate
Since the coefficients of the polynomial are real, if is a zero, its complex conjugate is also a zero. Therefore, is another zero.
Step 2: Factorization Using Known Zeros
We know that and are zeros, so the polynomial can be divided by , which simplifies to .
Now, we can factor the polynomial by dividing by .
Step 3: Polynomial Division
We divide:
Performing the division step-by-step:
- Multiply
- Subtract:
- Multiply
- Subtract:
So, the quotient is , meaning:
Step 4: Find the Remaining Zero
The remaining factor is , so the remaining zero is .
Final Answer:
The zeros of the polynomial are:
Would you like more details or explanations about any step? Here are five related questions:
- How do you perform polynomial division in more complex cases?
- Why must the complex conjugate also be a zero if is?
- Can you explain the significance of real and complex zeros in polynomials?
- What happens if a polynomial has only real coefficients but a zero is complex?
- How does the degree of a polynomial relate to the number of its zeros?
Tip: When working with polynomials with real coefficients, always check if complex zeros occur in conjugate pairs.
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Math Problem Analysis
Mathematical Concepts
Polynomial Functions
Complex Numbers
Polynomial Division
Factorization
Formulas
Polynomial division formula: P(x) = (x^2 + 1)(x + 9)
Complex conjugate: If -i is a zero, i is also a zero
Theorems
Complex Conjugate Theorem
Fundamental Theorem of Algebra
Suitable Grade Level
Grades 10-12
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