Write a polynomial equation with integral coefficients whase
zeros are 3 and .

Question

Write a polynomial equation with integral coefficients whase zeros are 3 and $$ 2 i $$.

UpStudy AI · Accepted Answer

To construct a polynomial equation with integer coefficients that has zeros at $$ 3 $$ and $$ 2i $$, follow these steps:

1. **Include the Complex Conjugate:**
   Since the coefficients of the polynomial are integers (and hence real), the complex conjugate of $$ 2i $$, which is $$ -2i $$, must also be a zero of the polynomial.

2. **Form the Factors:**
   The zeros are $$ 3 $$, $$ 2i $$, and $$ -2i $$. Therefore, the factors corresponding to these zeros are:
   $$
   (x - 3), \quad (x - 2i), \quad \text{and} \quad (x + 2i)
   $$

3. **Multiply the Factors:**
   Multiply the factors to obtain the polynomial:
   $$
   (x - 3)(x - 2i)(x + 2i)
   $$

4. **Simplify the Polynomial:**
   First, multiply the complex factors:
   $$
   (x - 2i)(x + 2i) = x^2 + (2i - 2i)x - (2i)^2 = x^2 + 4
   $$
   Then multiply by the remaining factor:
   $$
   (x - 3)(x^2 + 4) = x^3 + 4x - 3x^2 - 12
   $$
   Rearranging the terms gives:
   $$
   x^3 - 3x^2 + 4x - 12 = 0
   $$

**Final Polynomial Equation:**
$$
x^3 - 3x^2 + 4x - 12 = 0
$$

Answer: After simplifying, one such equation is
 x³ – 3 x² + 4 x – 12 = 0.
One polynomial equation with integer coefficients that has zeros at 3 and $$ 2i $$ is: $$ x^3 - 3x^2 + 4x - 12 = 0 $$

Write a polynomial equation with integral coefficients whase
zeros are 3 and .

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Write a polynomial equation with integral coefficients whase zeros are 3 and .

Solución de inteligencia artificial de Upstudy

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Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions

Write a polynomial equation with integral coefficients whase
zeros are 3 and .