PRELIMINARY ALGEBRA
at a value ofxfor whichφ(x) is also zero. Then the graph ofφ(x)justtouches
thex-axis. When this happens the value ofxso found is, in fact, a double real
root of the polynomial equation (corresponding to one of themkin (1.9) having
the value 2) and must be counted twice when determining the number of real
roots.
Finally, then, we are in a position to decide the number of real roots of theequation
g(x)=4x^3 +3x^2 − 6 x−1=0.The equationg′(x)=0,withg′(x)=12x^2 +6x−6, is a quadratic equation with
explicit solutions§
β 1 , 2 =− 3 ±√
9+72
12,so thatβ 1 =−1andβ 2 =^12. The corresponding values ofg(x)areg(β 1 )=4and
g(β 2 )=−^114 , which are of opposite sign. This indicates that 4x^3 +3x^2 − 6 x−1=0
has three real roots, one lying in the range− 1 <x<^12 and the others one on
each side of that range.
The techniques we have developed above have been used to tackle a cubicequation, but they can be applied to polynomial equationsf(x) = 0 of degree
greater than 3. However, much of the analysis centres around the equation
f′(x) = 0 and this itself, being then a polynomial equation of degree 3 or more,
either has no closed-form general solution or one that is complicated to evaluate.
Thus the amount of information that can be obtained about the roots off(x)=0
is correspondingly reduced.
A more general caseTo illustrate what can (and cannot) be done in the more general case we now
investigate as far as possible the real roots of
f(x)=x^7 +5x^6 +x^4 −x^3 +x^2 −2=0.The following points can be made.
(i) This is a seventh-degree polynomial equation; therefore the number of
realrootsis1,3,5or7.
(ii)f(0) is negative whilstf(∞)=+∞, so there must be at least one positive
root.§The two rootsβ 1 ,β 2 are written asβ 1 , 2. By conventionβ 1 refers to the upper symbol in±,β 2 to
the lower symbol.