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not be found analytically. MATLAB has a built-in command that finds all the
roots (real and complex) for any polynomial equation. As previously noted,
the MATLAB command for finding the polynomial roots is roots:
r=roots(p)

In interpreting the results from this command, recall the Fundamental Theo-
rem of Algebra, which states the root properties of a polynomial of degree n
with real coefficients:

1. The n polynomial admits n complex roots.
th
2. Complex roots come in conjugate pairs. [If you are not familiar
with complex numbers and with the term complex conjugate (the
latter term should pique your curiosity), be a little patient. Help is
on the way; Chapter 6 covers the topic of complex numbers].

Inversely, knowing the roots, we can reassemble the polynomial. The com-
mand is poly.

poly(r)

In-Class Exercise

Pb. 5.14 Find the roots of the polynomial p = [13210 3], and com-
pute their sum and product.
Pb. 5.15 Consider the two polynomials:

p = [132103]and p = [3 2 1]
1
2
Find the value(s) of x at which the curves representing these polynomials
would intersect.
Pb. 5.16 Find the constants A, B, C, D, and a, b, c, d that permits the follow-
ing expansion in partial fractions:

1 = A + B + C + D
x − 25x + 144 ( x − a) ( xb− ) ( xc− ) ( xd− )
4
2

5.3 Optimization Methods

Many design problems call for the maximization or minimization (optimiza-
tion) of a particular function belonging to a particular domain. (Recall the

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