**Introduction to One-Variable, Higher-Order Equations**

As the exponents in single-variable equations get bigger, finding the solutions becomes a more challenging business. In the olden days, a lot of insight, guesswork, and tedium was involved in solving such equations. Today, scientists have the help of computers, and when problems are encountered containing equations with variables raised to large powers, they just let a computer take over. The material here is presented so you won’t be taken aback if you ever come across *one-variable, higher-order equations* .

**The Cubic**

A cubic equation, also called a *one-variable, third-order equation* or a *third-order equation in one variable,* can be written in the following standard form:

*ax* ^{3} + *bx* ^{2} + *cx* + *d* = 0

where *a* , *b* , *c* , and *d* are constants, *x* is the variable, and *a* ≠ 0.

If you’re lucky, you’ll be able to reduce a cubic equation to factored form to find real-number solutions – *r* , – *s* , and – *t* :

( *x* + *r* )( *x* + *s* )( *x* + *t* ) = 0

Don’t count on being able to factor a cubic equation. Sometimes it’s easy, but most of the time it is nigh impossible. There is a *cubic formula* that can be used in a manner similar to the way in which the quadratic formula is used for quadratic equations, but it’s complicated, and is beyond the scope of this discussion.

**Plotting Cubics**

When we substitute *y* for 0 in the standard form of a cubic equation and then graph the resulting relation with *x* on the horizontal axis and *y* on the vertical axis, a curve with a characteristic shape results.

Figure 6-5 is a graph of the simplest possible cubic equation:

*x* ^{3} = *y*

**Fig. 6-5.** Graph of the cubic equation *x* ^{3} = *y* .

**Fig. 6-6.** Graph of the cubic equation (–1/2) *x* ^{3} = *y* .

The domain and range of this function both encompass the entire set of real numbers. This makes the cubic curve different from the parabola, whose range is always limited to only a portion of the set of real numbers. Figure 6-6 is a graph of another simple cubic:

(–1/2) *x* ^{3} = *y*

In this case, the domain and range also span all the real numbers.

The *inflection point,* or the place where the curve goes from concave downward to concave upward, is at the origin (0, 0) in both Fig. 6-5 and Fig. 6-6. This is because the coefficients *b* , *c* , and *d* are all equal to 0. When these coefficients are nonzero, the inflection point is not necessarily at the origin.

**The Quartic**

A *quartic equation,* also called a *one-variable, fourth-order equation* or a *fourth-order equation in one variable,* can be written in the following standard form:

*ax* ^{4} + *bx* ^{3} + *cx* ^{2} + *dx* + *e* = 0

where *a* , *b* , *c* , *d* , and *e* are constants, *x* is the variable, and *a* ≠ 0.

Once in a while you will be able to reduce a quartic equation to factored form to find real-number solutions – *r* , – *s* , – *t* , and – *u* :

( *x* + *r* )( *x* + *s* )( *x* + *t* )( *x* + *u* ) = 0

As is the case with the cubic, you will be lucky if you can factor a quartic equation into this form and thus find four real-number solutions with ease.

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