perfect square). For example, to solve the equation x^2 3 x4 by factoring, write the
equation in standard quadratic equation form: x^2 3 x 4 0. Then, factor the
form: (x4)(x1) 0.
In order for these numbers to equal zero, we determine that for (x4), xwould
be 4; and for (x1), xwould equal 1. Thus, the solutions are 4 and 1.
Can all quadratic equationsbe solved by factoring?
Don’t be fooled: Not all quadratic equations can be solved by factoring. For example,
x^2 3 x3 is not solvable with this method. One way to solve quadratic equations is
by completing the square; still another method is to graph the solution (a quadratic
graph forms a parabola—a U-shaped line seen on the graph). But one of the most well-
known ways is by using the quadratic formula.
For example, if we want to find the roots of the polynomial x^2 2 x7, we can
replace the “corresponding” numbers from the initial equation into the quadratic
equation ax^2 bx c 0. Thus, a1, b2, and c7. Substituting these num-
bers into the quadratic formula, we solve for:
bac 2 a (^4) and bac 2 a 4
-+ -bb^22 -- -
155
ALGEBRA
What is the Fundamental Theorem of Algebra?
T
he Fundamental Theorem of Algebra (FTA) is nothing new; it was first proved
by mathematician Carl Friedrich Gauss (1777–1855) in 1799. The equation
was as follows:anxnan 1 xn^1 ... a 1 x^1 a 0 0 (as long as nis greater than or equal
to 1 and anis not zero, and has at least one root in the complex numbers).The proof of this theorem goes on for pages—far beyond the scope of this
book. What all those proofs, numbers, and letters boil down to is that a polyno-
mial equation must have at least one number in its solution. It also tells us when
we have factored a polynomial completely. Simple enough, but like much of
mathematics, someone had to prove it.
But that is not all: The FTA is not constructive, and therefore it does not tell
us how to completely factor a polynomial. In other words, in reality, no one real-
ly knows how to factor a polynomial; we only know how to apply techniques to
certain kinds of polynomials. In fact, French mathematician Evariste Galois
(1811–1832), who died tragically in a duel, proved that there will never be a gen-
eral formula that will solve fifth degree or higher polynomials.