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Finding the roots of an equation is one of the key tasks in solving polynomial equations. The roots, sometimes referred to as solutions or zeros, are the values for which the equation equals zero. For instance, in an equation such as \(x^3 + x + 1 = 0\), the roots are the values of \(x\) that make the equation true.

Understanding the concept of roots is crucial because they represent the points where the graph of the polynomial crosses or touches the x-axis. Each root tells us where this crossing or touching occurs.

• Real roots appear on the x-axis as actual crossing points.
• Complex roots appear in conjugate pairs and do not cross the x-axis.

If an equation has double roots, these are roots that repeat, causing the graph to only touch the x-axis at those points, rather than crossing.

The substitution method is a strategic tool used to simplify complex equations for easier solving. It often involves replacing a variable with another variable or expression to help transform the equation.

In the given problem, we use substitution to find an equation whose roots are double the original equation's roots. By letting \(x = 2y\), we substitute this into the original equation \(x^3 + x + 1 = 0\).

This replacement transforms the equation into a new form:

• Start with \(x = 2y\)
• Replace the \(x\) in the equation with \(2y\)
• The equation \((2y)^3 + 2y + 1 = 0\) then emerges

The substitution method helps to frame a new equation while maintaining the relationship between the roots of the original and transformed equations.

Transforming an equation means altering its form without changing its fundamental solutions or characteristics. This process can involve a variety of algebraic manipulations such as expanding, factoring, or simplifying.

In the problem at hand, the transformation involves first substituting \(x\) with \(2y\). After substitution, simplifying the equation yields \(8y^3 + 2y + 1 = 0\).

Transformation steps include:

• Changing the original variable \(x\) to \(2y\), reflecting a scaling of the roots.
• Raising \(2y\) to the power of 3 to address the cubic term.
• Expanding terms like \((2y)^3\) to get \(8y^3\).
• Adding the constant terms as needed to reflect the new variable.

Equation transformation is key in understanding how operations like scaling or shifting roots impact the overall equation. It ensures that the new equation is equivalent in solution to the original but with the desired characteristics, such as having roots that are twice as large.