College Algebra 9th txtbk

288 CHAPTER 4 POLYNOMIAL AND RATIONAL FUNCTIONS
(see the figure). Find the value of x that would result in a box with
a volume of 500 cubic inches.
20 in.
x
x
40 in.
76. SHIPPING A shipping box is reinforced with steel bands in all
three directions (see the figure). A total of 20.5 feet of steel tape is
to be used, with 6 inches of waste because of a 2-inch overlap in
each direction. If the box has a square base and a volume of 2 cubic
feet, find the side length of the base.
75. CONSTRUCTION A propane gas tank is in the shape of a right
circular cylinder with a hemisphere at each end (see the figure). If
the overall length of the tank is 10 feet and the volume is 20 cubic
feet, find the common radius of the hemispheres and the cylinder.
x
x
y
x
x
10 feet
4-3
Complex Zeros and Rational Zeros
of Polynomials
Z The Fundamental Theorem of Algebra
Z Factors of Polynomials with Real Coefficients
Z Graphs of Polynomials with Real Coefficients
Z Rational Zeros
The graph of the polynomial function P(x) x 2 4 does not cross the x axis, so P(x) has
no real zeros. It does, however, have complex zeros, 2i and 2i; by the factor theorem,
x 2 4 (x 2i)(x 2i). The fundamental theorem of algebra guarantees that every nonconstant
polynomial with real or complex coefficients has a complex zero; as a result, such
a polynomial can be factored as a product of linear factors. In Section 4-3, we study the
fundamental theorem and its implications, including results on the graphs of polynomials
with real coefficients. Finally, we consider a problem that has led to important advances in
mathematics and its applications: When can zeros of a polynomial be found exactly?
Z The Fundamental Theorem of Algebra
The fundamental theorem of algebra was proved by Karl Friedrich Gauss (1777–1855), one
of the greatest mathematicians of all time, in his doctoral thesis. A proof of the theorem is
beyond the scope of this book, so we will state and use it without proof.