Chapter 17
CALCULATING ROOTS OF FUNCTIONS
Chapter 7 was dedicated primarily to the solution of linear equations (equa
tions with degree one) and Chapter 14 almost entirely focused on the solu
tion of quadratic equations (which have degree two), so it is moderately
surprising that equations (and functions) of degree three and higher do
not each require separate solution techniques. Instead, solutions can be
determined analytically by means of a common set of theorems, including
the Fundamental Theorem of Algebra, the remainder theorem, the leading
coefﬁcient test, Descartes’ rule of signs, and the rational root test.
There might be a quadratic formula, but there’s no cubic formula that
solves cubic equations using only their coefcients—the same goes for
equations of degree four, degree ve, and so on. One thing you should know:
Instead of calculating the solutions of equations, most of the time you calculate
the roots of functions. Essentially, it means the same thing—you just use
different words.
Transforming an equation into a function is simple—just solve the equation for 0.
For example, to solve the equation x
2
= 9, subtract 9 from both sides (to get
x
2
– 9 = 0), and instead of writing 0, give the function a name (so you end up
with f(x) = x
2
– 9). The solutions of the equation x
2
= 9 and the roots of the
function f(x) = x
2
– 9 are equal: x = –3 or x = 3.
To calculate the roots of functions, use a combination of different tests to
learn about the function and its graph. Do its ends point in the same direction,
or do they go different directions, like one going up and one going down? How
many positive roots are there? How many are negative? How in the world do you
factor a polynomial like x
3
+ 13x
2
+ 31x – 45?
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