Question

In the past, you’ve seen linear and quadratic equations and their graphs. A linear equation is a polynomial of order 1, which means that the greatest exponent for a variable is 1. A quadratic equation is a polynomial of order 2. In what types of situations could you make use of polynomials of order 3 or higher? How would the graphs of these polynomials change? How would they change if there was a variable in a denominator of an equation? What if the variable was part of an exponent?

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Answer 1

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An equation with degree three is called a cubic equation. The nature of roots of all cubic equations is either one real root and two imaginary roots or three real roots. If the polynomials have degree three, they are known as cubic polynomials.

A cubic polynomial function of the third degree has the form shown on the right and it can be represented as y = ax3 + bx2 + cx + d, where a, b, c, and d are real numbers and a ≠ 0.

To solve an equation with a variable in a fraciton, treat the denominator as a constant value and multiply both sides of the equation by the denominator in order to eliminate it.

Rule 1: a1/m × a1/n = a. (1/m + 1/n)

Rule 2: a1/m ÷ a1/n = a. (1/m - 1/n)

Rule 3: a1/m × b1/m = (ab) 1/m

Rule 4: a1/m ÷ b1/m = (a÷b) 1/m

Rule 5: a-m/n = (1/a) m/n