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A cubic polynomial is a type of polynomial that is characterized by having a degree of three. This means the highest power of the variable, usually denoted as \(x\), is three. The general form of a cubic polynomial is \(p(x) = ax^3 + bx^2 + cx + d\). Here:

• \(a\), \(b\), \(c\), and \(d\) are constants known as coefficients.
• \(x^3\) is the term that gives the polynomial its cubic nature, determining the polynomial's highest degree.
• If \(a\) is non-zero, then the polynomial maintains its cubic form.

Cubic polynomials can curve more dynamically compared to linear or quadratic polynomials, which means they can pass through any set of four different points on a 2D plane, as long as the points allow it. This makes cubic polynomials incredibly useful in interpolation, where you need a curve to fit exactly through multiple points. The curvature and flexibility they offer are ideal for such tasks.

A system of equations is essentially a set of two or more equations that we deal with together. In the context of finding a cubic polynomial, we use a system of equations to determine the polynomial coefficients that will make the polynomial curve go through the given points. Each point provides a specific equation when plugged into the polynomial formula. For instance, given the original points:

• \((0,0)\) leads to the equation \(d = 0\).
• \((1,2)\) gives us \(a + b + c + d = 2\).
• \((4,-3)\) provides \(64a + 16b + 4c + d = -3\).
• \((10,-1)\) results in \(1000a + 100b + 10c + d = -1\).

With \(d\) determined as zero from the point \((0,0)\), the system becomes a simplified set of three equations that involves only \(a\), \(b\), and \(c\). Solving this system involves finding a common solution that satisfies all equations. Various methods like substitution, elimination, or using matrices can be employed to find the values of \(a\), \(b\), and \(c\). Solving systems of equations is crucial for uniquely defining the cubic polynomial that meets the problem's requirements.

The coefficients of a polynomial are crucial as they dictate the polynomial's shape and position. For a cubic polynomial in the form \(p(x) = ax^3 + bx^2 + cx + d\), the coefficients \(a\), \(b\), \(c\), and \(d\) play significant roles:

• \(a\) influences the leading behavior. A positive \(a\) will make the curve rise to the right, while a negative \(a\) causes it to fall.
• \(b\) and \(c\) account for how the curve twists and bends.
• \(d\), the constant term, shifts the curve up or down without affecting its shape.

In this exercise, solving the system of equations provided the values \(a = \frac{7}{60}\), \(b = -\frac{23}{60}\), \(c = \frac{41}{30}\), and \(d = 0\). These coefficients ensure that our polynomial fits exactly through the specified points \((0,0), (1,2), (4,-3), (10,-1)\). Calculating the coefficients typically involves some algebraic manipulation and, as such, is a key step in the interpolation process, which validates the precision of cubic polynomials in fitting data points.