Question

What is the equation of a quadratic function P with rational coefficients that

has a zero of 3 + 7i?

Answer 1

Answer:

$P(x) = x^2-6x+58$

Step-by-step explanation:

Quadratic function is function of the form $P(x)=ax^2+bx+c$ where a, b and c are real numbers and $a\neq 0$

A number is said to be a rational number if it can be written in the form $\frac{u}{v}$ where u and v are integers and $v\neq 0$.

A number is said to be a complex number if it can be written in the form u+iv where u and v are real numbers.

A number 'a' is said to be a zero of a function P(x) if $P(a)=0$

We know that if zero of a fraction is of form u+iv then its other zero is u-iv.

As 3+7i is a zero of the function, 3-7i is also its zero.

So, given equation is $\left ( x-(3-7i) \right )\left ( x-(3+7i) \right )$

$P(x)=\left ( x-(3-7i) \right )\left ( x-(3+7i) \right )\\=x^2-x(3+7i)-x(3-7i)+(3-7i)(3+7i)\\=x^2-3x-7ix-3x+7ix+9+49\\=x^2-6x+58$

Answer 2

The equation of a quadratic function P with rational coefficients that have a zero of 3 + 7i is;

P(x) = x² - 6x + 58

We are given one of the zero of the quadratic function as 3 + 7i.

The given zero of the function is a complex number and we know that when dealing with complex numbers, if the zero of the function is of the form a + bi then we can deduce that the other zero of the function will be a - bi.

This means that the other zero of our function is 3 - 7i.

Thus,the factors of the function are;

(x - (3 + 7i)) and (x - (3 - 7i))

Thus, the equation of the quadratic function is;

P(x) = (x - (3 + 7i)) × (x - (3 - 7i))

We know that i² = -1

Thus, our equation is;

P(x) = x² - 6x + 58

Read more about quadratic equation with complex roots at; https://brainly.com/question/11371661