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The x-intercepts of a polynomial function are the points where the function crosses the x-axis. This occurs when the output or value of the function, denoted as \(f(x)\), equals zero. To find the x-intercepts, set the function itself to zero and solve for \(x\).
For the function given in the exercise, \(f(x)=(x-4)(2x^2-2x+1)\), the x-intercept can be easily identified by focusing on the factor \(x-4\). Solving \(x-4=0\) gives \(x=4\), so there is an x-intercept at \(x=4\).
The second part of the function, \(2x^2-2x+1\), is a quadratic equation. To find any additional x-intercepts from this part, use the quadratic formula: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).
However, upon calculating the discriminant \(b^2-4ac\) in this specific case, it is less than zero. A negative discriminant means there are no real roots, implying no x-intercepts from this factor. Thus, the only x-intercept is at \(x=4\).
Understanding x-intercepts is crucial because they provide insight into where the function changes sign and how it interacts with the x-axis.

A y-intercept is where the graph of a function crosses the y-axis. This point gives a clear indication of the starting value of the function when \(x=0\). To find the y-intercept, substitute \(x=0\) into the function, simplifying to find the value of \(f(x)\).
In the given function \(f(x)=(x-4)(2x^2-2x+1)\), by substituting ~\(x=0\), we have:

• \(f(0) = (0-4)(2(0)^2 - 2(0) + 1)\)
• \(f(0) = (-4)(1) = -4\)

Therefore, the y-intercept is at \(y=-4\). This point \((0, -4)\) is essential for sketching the graph, as it tells us where the curve starts vertically when \(x\) is zero. Recognizing y-intercepts helps in building an understanding of the vertical position of a polynomial function on a graph.

The extremities behavior of a polynomial function describes how the function behaves as \(x\) approaches positive or negative infinity. It helps us visualize how the graph stretches or compresses towards the ends.
Degree and leading coefficient are two key factors influencing extremities behavior:

• The degree of our polynomial is 3 (since it's a third-degree polynomial).
• The leading coefficient, which is the coefficient of the term with the highest degree, is positive.

These characteristics allow us to predict the extremities behavior:

• As \(x\to -\infty\), \(f(x) \to -\infty\). This means the graph stretches downward on the left side.
• As \(x\to \infty\), \(f(x) \to \infty\). So, the graph stretches upward on the right side.

By analyzing the behavior at the extremities, students can better comprehend how the polynomial's graph unfolds across the x-axis. This understanding aids in creating a more accurate sketch of the function and anticipating curvature directions at the far left and right.