Chapter Seventeen — Calculating Roots of Functions
The Humongous Book of Algebra Problems
Identifying Rational Roots
Factoring polynomials given a head start
17.1 Given a polynomial of degree n that has real number coefﬁcients, what is
guaranteed by the Fundamental Theorem of Algebra?
The Fundamental Theorem of Algebra states that the polynomial will have
exactly n complex roots. It is an existence theorem, merely guaranteeing that
those roots exist but providing no means by which to calculate them. Neither
can the theorem be used to further classify the roots, such as to determine how
many of the complex roots are real numbers or how many are rational.
17.2 According to the remainder theorem, if the quotient (ax
+ cx + d) ÷
(x – m) has remainder r, what conclusion can be drawn? Assume that a, b, c, d,
and m are real numbers.
The remainder theorem states that substituting x = m into a polynomial written
in terms of x produces a value exactly equal to the remainder when that poly-
nomial is divided by x – m. In this example, dividing ax
+ cx + d by x – m
produces remainder r, so the remainder theorem states that substituting x = m
into the expression results in the same value.
+ c(m) + d = r
Note: Problems 17.3–17.4 refer to the polynomial x
+ 7x – 4.
17.3 Evaluate the expression for x = 5 using the remainder theorem. Verify your
According to the remainder theorem, the remainder of (x
+ 7x – 4) ÷
(x – 5) is equal to the expression x
+ 7x – 4 evaluated for x = 5. Calculate
the quotient using synthetic division.
+ 7x – 4 = 81 when x = 5. Verify this by actually substituting
x = 5 into the expression and simplifying.
written in terms
of x” is a polynomial
containing x’s and no
left over when
you divide a
(x – m) is equal to the
number you get out
when you plug
x = m into the
check out Problems