Brilliance Math Tutoring

📝👀 At first glance, we see perfect squares. This will make today’s exercise marginally easier. In step 1, “x - 1” is found to be a factor after some trial-and-error. Trial-and-error meaning substitute “1” for “x” in f(x) and verify whether or not f(x = 1) = 0. If true, “x - 1” is a factor of the given polynomial. If false, “x - 1” is not a factor. Proceed by testing 2, then 3, then 4, and so on. (The same can be done with negative numbers).

Next, we separate “x - 1” into two ‘groups’ with each group being multiplied by a distinct coefficient. The sum of these coefficients is in fact the quotient of “4(x^3) - 4(x^2) - 25x + 25” divided by “x - 1.” Further factorization of the said quotient, gives us two additional factors: “2x + 5” and “2x - 5.” Solving each of the three factors results in the x-values shown in step 1.

Step 2 illustrates a derivation, f’(x), and domains where f(x) increases and decreases.

Step 3 covers an additional derivation, f”(x), and domains where f(x) “opens up” and “opens down.”

🚨 Given: Scores from five (5) random students.

Task: Approximate the linear function that describes the relation between the points.

Solution: Use Linear Regression to approximate the line that best represents the given data, where "b" is the slope of the line, and "a" is the y-intercept in the following formula.

y = bX + a