Chapter Twenty — Rational Expressions
The Humongous Book of Algebra Problems
451
Note: Problems 20.18–20.19 refer to the function f(x) deﬁned below.
20.19 If , given function d(x) deﬁned in Problem 20.18,
calculate the values of a, b, and c.
According to Problem 20.18, .
The problem states that .
Set the rational functions equal.
Two equivalent fractions with equivalent denominators must have equivalent
numerators.
4x
3
+ 7x
2
+ 37x – 103 = ax
3
+ (a + b)x
2
+ (1 + 9b – c)x + (–31 + 8c)
For the cubic polynomials to be equal, each of the corresponding terms must be
equal.
Set the x
3
terms equal and solve for a.
Set the x
2
terms equal, substitute a = 4 into the equation, and solve for b.
Set the xterms equal, substitute b = 3 into the equation, and solve for c.
Because f(x)
equals both of the
fractions (the one
that starts with ax
3
in
the numerator and the
one that starts with
4x
3
), those fractions are
equal to each other.
You can drop the
denominators and
set the numerators
equal.
In other
words, the x
3

terms are equal
(4x
3
= ax
3
), the x
2

terms are equal
(7x
2
= [a + b]x
2
), the
x–terms are equal,
and the constants
are equal.