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:

P(x)=x3+9x2+x+9P(x) = x^3 + 9x^2 + x + 9

and one of its zeros, x=ix = -i.

Step 1: Complex Conjugate

Since the coefficients of the polynomial are real, if i-i is a zero, its complex conjugate ii is also a zero. Therefore, x=ix = i is another zero.

Step 2: Factorization Using Known Zeros

We know that x=ix = -i and x=ix = i are zeros, so the polynomial can be divided by (x+i)(xi)(x + i)(x - i), which simplifies to x2+1x^2 + 1.

Now, we can factor the polynomial by dividing P(x)P(x) by x2+1x^2 + 1.

Step 3: Polynomial Division

We divide:

x3+9x2+x+9x2+1\frac{x^3 + 9x^2 + x + 9}{x^2 + 1}

Performing the division step-by-step:

  1. x3÷x2=xx^3 \div x^2 = x
  2. Multiply x×(x2+1)=x3+xx \times (x^2 + 1) = x^3 + x
  3. Subtract: (x3+9x2+x+9)(x3+x)=9x2+9(x^3 + 9x^2 + x + 9) - (x^3 + x) = 9x^2 + 9
  4. 9x2÷x2=99x^2 \div x^2 = 9
  5. Multiply 9×(x2+1)=9x2+99 \times (x^2 + 1) = 9x^2 + 9
  6. Subtract: (9x2+9)(9x2+9)=0(9x^2 + 9) - (9x^2 + 9) = 0

So, the quotient is x+9x + 9, meaning:

P(x)=(x2+1)(x+9)P(x) = (x^2 + 1)(x + 9)

Step 4: Find the Remaining Zero

The remaining factor is x+9x + 9, so the remaining zero is x=9x = -9.

Final Answer:

The zeros of the polynomial are: x=i,i,9x = -i, i, -9

Would you like more details or explanations about any step? Here are five related questions:

  1. How do you perform polynomial division in more complex cases?
  2. Why must the complex conjugate ii also be a zero if i-i is?
  3. Can you explain the significance of real and complex zeros in polynomials?
  4. What happens if a polynomial has only real coefficients but a zero is complex?
  5. 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