Math Problem Statement

Find the complex zeros of the polynomial function f(x) = x^3 + 8 and write f in factored form.

Solution

To solve this problem, we need to express the polynomial f(x)=x3+8f(x) = x^3 + 8 in factored form using its complex zeros. Here’s the step-by-step solution:

Step 1: Recognize the Complex Zeros

The complex zeros of f(x)=x3+8f(x) = x^3 + 8 are given as: 2,1+i3,1i3-2, \quad 1 + i\sqrt{3}, \quad 1 - i\sqrt{3} These are the roots of the polynomial.

Step 2: Write f(x)f(x) in Factored Form

Since we know the roots, we can write f(x)f(x) as the product of terms of the form (xroot)(x - \text{root}) for each root: f(x)=(x+2)(x(1+i3))(x(1i3))f(x) = (x + 2)(x - (1 + i\sqrt{3}))(x - (1 - i\sqrt{3}))

Step 3: Simplify the Complex Conjugate Terms

Notice that (x(1+i3))(x(1i3))(x - (1 + i\sqrt{3}))(x - (1 - i\sqrt{3})) involves a pair of complex conjugates, which we can simplify as follows: (x(1+i3))(x(1i3))=(x1)2(i3)2(x - (1 + i\sqrt{3}))(x - (1 - i\sqrt{3})) = \left( x - 1 \right)^2 - \left( i\sqrt{3} \right)^2 Since i2=1i^2 = -1, we get: =(x1)2(3)=(x1)2+3= (x - 1)^2 - (-3) = (x - 1)^2 + 3 Expanding (x1)2+3(x - 1)^2 + 3: =x22x+1+3=x22x+4= x^2 - 2x + 1 + 3 = x^2 - 2x + 4

Step 4: Write the Final Factored Form

Now we can express f(x)f(x) completely in factored form: f(x)=(x+2)(x22x+4)f(x) = (x + 2)(x^2 - 2x + 4)

This is the complete factorization of f(x)=x3+8f(x) = x^3 + 8 using its complex zeros.

Would you like more details or have any questions?


Here are five related questions to expand on this problem:

  1. How do you factor other polynomials with real and complex roots?
  2. What is the significance of complex conjugate pairs in polynomial factorization?
  3. How do you derive the complex roots of a cubic equation like x3+8=0x^3 + 8 = 0?
  4. Why is it useful to express a polynomial in its factored form?
  5. 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.

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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