Question

Let \(\alpha\) and \(\beta\) be the roots of the equation \(x^{2}+x+1=0\). The equation whose roots are \(\alpha^{19}, \beta^{7}\) is : (a) \(x^{2}-x-1=0\) (b) \(x^{2}-x+1=0\) (c) \(x^{2}+x-1=0\) (d) \(x^{2}+x+1=0\)

Short answer

Question: Given that the roots of the equation \(x^2 + x + 1 = 0\) are \(\alpha\) and \(\beta\), find the equation having roots equal to \(\alpha^{19}\) and \(\beta^7\). Answer: The equation having roots equal to \(\alpha^{19}\) and \(\beta^7\) is (c) \(x^{2}+x-1=0\).

Step-by-step solution

Rewrite the equation in terms of its roots

The given equation is \(x^{2}+x+1=0\). It has two roots, \(\alpha\) and \(\beta\). We know that the sum of the roots is equal to the opposite of the coefficient of the \(x\) term and the product of the roots is equal to the constant term. Since the coefficients are all 1, we can write the following two relations: \(\alpha+\beta = -1\) (sum of roots) and \(\alpha \cdot \beta = 1\) (product of roots).

Find a relationship between the new roots

We are asked to find the equation with roots \(\alpha^{19}\) and \(\beta^{7}\). Since we are given an equation with roots \(\alpha\) and \(\beta\), we can find the sum and product of these new roots using the relations from Step 1. Denote the sum as \(S\): \(S = \alpha^{19} + \beta^{7}\) Denote the product as \(P\): \(P = \alpha^{19} \cdot \beta^{7}\)

Simplify the sum and product of the new roots

We already have the sum \(S\) as \(\alpha^{19} + \beta^{7}\). In order to find the product \(P\), we can use the relation for the product of the roots of the original equation. According to this relation, \(\alpha \cdot \beta = -1\). From here, we can raise both sides to the powers that will make \(P\), which in this case are the lcm of 19 and 7: \(133\). \(\left(\alpha \cdot \beta\right)^{133} = (-1)^{133}\) \(\Rightarrow \alpha^{133} \cdot \beta^{133} = -1\). Using this relation, we can find the value of \(P\): \(P = \alpha^{19} \cdot \beta^{7} = \alpha^{126} \cdot \beta^{126} \cdot \alpha^{133} \cdot \beta^{133}\) \(\Rightarrow P = (\alpha^{126} \cdot \beta^{126}) \cdot (\alpha^{133} \cdot \beta^{133}) = \left(\alpha \cdot \beta\right)^{126} \cdot \left(\alpha \cdot \beta\right)^{133} = (-1)^{126} \cdot (-1)^{7} = -1\)

Find the equation corresponding to the new roots

Now that we have the sum and product of the new roots \(\alpha^{19}\) and \(\beta^{7}\), we can write the equation corresponding to these roots using the relation between the roots and the coefficients of the equation. The equation will be of the form \(x^2 - Sx + P = 0\). Plugging in the values for \(S\) and \(P\), we get the equation: \(x^2 - (\alpha^{19} + \beta^{7})x - 1 = 0\)

Identify the answer among the given options

Comparing the equation from Step 4 with the given options, we see that the equation corresponds to option (c) \(x^{2}+x-1=0\). Thus, the correct answer is (c) \(x^{2}+x-1=0\).

Key concepts

Roots of Quadratic Equation

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The roots of a quadratic equation are the values of \(x\) that make the equation true. These roots can often be found using the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

For the equation \(x^2 + x + 1 = 0\), the terms are such that \(a = 1\), \(b = 1\), and \(c = 1\). This equation is particularly interesting because it involves complex roots, which occur when the discriminant \(b^2 - 4ac\) is negative. Here, the discriminant is \(-3\), indicating complex roots.
Using relationships between coefficients and roots, we get:

• Sum of the roots \(\alpha + \beta = -b/a = -1\)
• Product of the roots \(\alpha \cdot \beta = c/a = 1\)

Sum and Product of Roots

When dealing with quadratic equations, finding the sum and product of roots provides valuable insights. For a quadratic equation \(ax^2 + bx + c = 0\), if \(\alpha\) and \(\beta\) are roots, the following relationships hold:

• Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
• Product of roots: \(\alpha \cdot \beta = \frac{c}{a}\)

These formulas come from expanding \((x - \alpha)(x - \beta)\) and matching it to the structure of the quadratic equation.

Applying these to our given equation \(x^2 + x + 1 = 0\), we find \(\alpha + \beta = -1\) and \(\alpha \cdot \beta = 1\). These relationships set the stage for finding further powers of roots, such as \(\alpha^{19}\) and \(\beta^{7}\), by using known algebraic properties of roots and powers.

Higher Degree Powers of Roots

Calculating higher powers of roots \(\alpha\) and \(\beta\), like \(\alpha^{19}\) and \(\beta^{7}\), requires leveraging the initial relationships of the roots. Earlier, we established that \(\alpha \cdot \beta = 1\). This fact can be elevated to higher powers by observing:

• \((\alpha \cdot \beta)^n = (\alpha^n \cdot \beta^n)\)

Using these powers, we aim to construct new equations whose roots are the high powers of \(\alpha\) and \(\beta\). In our example, we computed the product \(\alpha^{19} \cdot \beta^{7}\), leading to \((-1)^7 = -1\).
Ultimately, these calculations helped derive a new quadratic equation \(x^2 + x - 1 = 0\), characterizing the powers of the original roots in the form of a standard quadratic equation.