Question

Let \(p\) and \(q\) be real numbers such that \(p \neq 0, p^{3} \neq q\) and \(p^{3} \neq-q .\) If \(\alpha\) and \(\beta\) are nonzero complex numbers satisfying \(\alpha+\beta=-\mathrm{p}\) and \(\alpha^{3}+\beta^{3}=\mathrm{q}\), then a quadratic equation having \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) as its roots is A) \(\left(p^{3}+q\right) x^{2}-\left(p^{3}+2 q\right) x+\left(p^{3}+q\right)=0\) B) \(\left(p^{3}+q\right) x^{2}-\left(p^{3}-2 q\right) x+\left(p^{3}+q\right)=0\) C) \(\left(p^{3}-q\right) x^{2}-\left(5 p^{3}-2 q\right) x+\left(p^{3}-q\right)=0\) D) \(\left(p^{3}-q\right) x^{2}-\left(5 p^{3}+2 q\right) x+\left(p^{3}-q\right)=0\)

Short answer

The correct quadratic equation is B) \(\left(p^{3}+q\right) x^{2}-\left(p^{3}-2 q\right) x+\left(p^{3}+q\right)=0\).

Step-by-step solution

Express \(\alpha^3 + \beta^3\) in another form

Use the identity \(\alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2)\) to express \(\alpha^3 + \beta^3\) in terms of \(\alpha + \beta\) and \(\alpha\beta\).

Find the value of \(\alpha\beta\)

Since \(\alpha + \beta = -p\) and \(\alpha^3 + \beta^3 = q\), plug these into the identity to get \(q = (-p)(\alpha^2 - \alpha\beta + \beta^2)\). Then solve for \(\alpha\beta\).

Obtain the quadratic equation for \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\)

Use the relationship \(x = \frac{\alpha}{\beta}\) or \(x = \frac{\beta}{\alpha}\) to form the equation \(x \alpha = \beta\) or \(\alpha = x \beta\), and find the quadratic equation by eliminating \(\alpha\) and \(\beta\).

Express the quadratic equation in terms of \(p\) and \(q\)

Substitute the value of \(\alpha\beta\) into the quadratic equation and simplify to express the coefficients in terms of \(p\) and \(q\).

Key concepts

Complex Numbers and Polynomials

When it comes to complex numbers, we're dealing with values that incorporate both a real part and an imaginary part. In a polynomial equation, complex numbers arise when we're solving for roots that are not real numbers. For instance, roots that involve the square root of a negative number, such as \( i \), where \( i \sqrt{-1} \). Imaginary numbers, when combined with real numbers, form complex numbers, which can be expressed as \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part.

In polynomials, especially quadratic equations, we consider these complex numbers for finding solutions when the discriminant (part under the square root in the quadratic formula \(b^2 - 4ac\)) is negative. This demonstrates that not all polynomials with real coefficients will have real solutions; they can have complex roots. Moreover, in accordance with the Fundamental Theorem of Algebra, every non-constant single-variable polynomial with complex coefficients has at least one complex root. This shows the deep connection between complex numbers and polynomial equations.

Quadratic Equation Roots

Quadratic equations are a special kind of polynomial with a degree of two and are generally expressed in the standard form \( ax^2 + bx + c = 0 \), where \( a \) is the coefficient of \( x^2 \) and is non-zero, and \( b \) and \( c \) are the coefficients of \( x \) and the constant term, respectively. The solutions or roots of a quadratic equation can be found using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). These roots are the values of \( x \) that satisfy the equation and can be real or complex numbers depending on the discriminant \( b^2 - 4ac \).

If the discriminant is positive, we have two distinct real roots. If it's zero, we have one real root, and if it’s negative, we have two complex roots which are conjugates of each other. Recognizing the nature of the roots is vital for understanding the solutions and the nature of the graph that represents the quadratic equation.

Sum and Product of Roots

The sum and product of roots of a quadratic equation holds a special place in understanding these equations. They are directly related to the coefficients of the quadratic equation. For an equation \( ax^2 + bx + c = 0 \), the sum of the roots, \( \alpha + \beta \) is given by \( -\frac{b}{a} \) and the product of the roots, \( \alpha\beta \) is \( \frac{c}{a} \). This information simplifies many problems related to quadratic equations, especially when we don’t need to determine the exact roots but need to understand their relationship or their combined effect.

For example, if we're given the sum and product of two roots, we can reconstruct the quadratic equation as \( x^2 - (\sum of roots)x + (product of roots) = 0 \). Utilizing the sum and product of roots can significantly simplify complex number operations within polynomials, especially when comparing properties of these roots without needing to solve them explicitly.