Find the Equation Given the Roots -i square root of 7
The problem is asking for the derivation of a polynomial equation based on the given roots or solutions of the equation. Specifically, the root provided is -i√7, which is a complex number. To find the equation, one would typically use the fact that if a complex number is a root of a real polynomial, then its conjugate is also a root. The question may be seeking a quadratic or a higher-degree polynomial for which -i√7 and its conjugate are solutions.
$- i \sqrt{7}$
Answer
Solution:
Step 1:
Identify the roots where the function crosses the x-axis, which occurs at $y = 0$.
Step 2:
The root $x = -i\sqrt{7}$ is determined by setting $x - (-i\sqrt{7}) = y$ to zero, resulting in the factor $x + i\sqrt{7}$.
Step 3:
Similarly, the root $x = i\sqrt{7}$ is found by setting $x - i\sqrt{7} = y$ to zero, yielding the factor $x - i\sqrt{7}$.
Step 4:
Form the equation by multiplying the factors: $y = (x + i\sqrt{7})(x - i\sqrt{7})$.
Step 5:
Expand the equation by multiplying the factors.
Step 5.1:
Use the FOIL method to expand $(x + i\sqrt{7})(x - i\sqrt{7})$.
Step 5.1.1:
Distribute the terms: $y = x(x - i\sqrt{7}) + i\sqrt{7}(x - i\sqrt{7})$.
Step 5.1.2:
Continue distribution: $y = x^2 - xi\sqrt{7} + i\sqrt{7}x - i^2(\sqrt{7})^2$.
Step 5.2:
Simplify the expression.
Step 5.2.1:
Combine like terms: $y = x^2 - xi\sqrt{7} + xi\sqrt{7} - i^2(\sqrt{7})^2$.
Step 5.2.2:
Simplify each term.
Step 5.2.2.1:
Calculate $x^2$: $y = x^2 - i^2(\sqrt{7})^2$.
Step 5.2.2.2:
Multiply $-i^2$ by $(\sqrt{7})^2$: $y = x^2 + (\sqrt{7})^2$.
Step 5.2.2.3:
Simplify the square of $\sqrt{7}$: $y = x^2 + 7$.
Step 6:
The final equation is $y = x^2 + 7$.
Knowledge Notes:
Roots of a Polynomial: The roots of a polynomial are the values of $x$ for which the polynomial equals zero. In the context of a quadratic equation, these are the points where the graph of the equation crosses the x-axis.
Complex Conjugates: If a polynomial has real coefficients and a complex root, the complex conjugate of that root is also a root. In this case, the roots are $-i\sqrt{7}$ and $i\sqrt{7}$, which are complex conjugates.
Multiplying Complex Numbers: When multiplying complex numbers, one should use the distributive property, also known as the FOIL method for binomials, to expand the product. The product of a complex number and its conjugate results in a real number.
Imaginary Unit: The imaginary unit $i$ is defined such that $i^2 = -1$. This property is used to simplify expressions involving $i$.
Squaring a Square Root: When a square root is squared, such as $(\sqrt{7})^2$, it simplifies to the number under the root, which is 7 in this case.
Simplifying Expressions: The process of simplifying expressions involves combining like terms, applying the distributive property, and reducing expressions to their simplest form.
Quadratic Equations: A quadratic equation in standard form is written as $y = ax^2 + bx + c$. When given the roots, the equation can be formed by multiplying the factors associated with each root.
