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In questions 231 to 235, for the given polynomial function p, (a) determine the
degree of p(x), (b) describe the domain and range of the graph of p, (c)
determine the zeros of p, (d) find the x-intercepts of the graph of p, and (e) find
the y-intercept of the graph of p. Refer to the following guidelines, as needed.
A polynomial function p such that
, where n is a nonnegative integer and an ≠ 0 is the leading coefficient, has
degree n, the highest exponent of x. A constant polynomial’s (p(x) = c, c ≠ 0)
degree is zero. The zero polynomial’s (p(x) = 0) degree is undefined. The
domain of p is R. If n is odd, the range is R; and if n is even, the range is a subset
of R. The zeros of p are the roots of the equation p(x) = 0. That is, r is a zero of p
if and only if p(r) = 0. If r is a real zero of p, then r is an x-intercept of the graph
of p. The y-intercept of the graph of p is p(0).
231. p(x) = 3(x – 1)(x 3)(x – 4)(x 2)(x – 2)
232. q(x) = (x
2 4)(x
2 – 5)(x
2 – 9)
233. g(x) = x
4 – 81
234. g(x) = –2x
2 – 9x 5
235. f(x) = 3x 5
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236. For the graph shown on the next page, identify (a) turning points and (b)
relative or absolute extrema. Refer to the following guidelines, as needed.
The graph of a polynomial function will have a turning point (x, y) whenever the
graph changes from increasing to decreasing or from decreasing to increasing.
The y-value of a turning point is either a relative maximum or relative minimum
value for the function. An nth-degree polynomial will have at most n – 1 turning
points.
Remainder Theorem, Factor Theorem, and Fundamental
Theorem of Algebra
237. Given p(x) = 2x
3 – 5x
2 – 14x 8, use the remainder theorem to find p(2).
238. Given p(x) = 2x
3 – 5x
2 – 14x 8, use the remainder theorem to find p(–2).
239. Use the factor theorem and the results in question 238 to factor p(x) = 2x
3 –
5x
2 – 14x 8 completely.
240. The zeros of a polynomial function p of degree 4 and leading coefficient 5
are 3, –2, and ± . Express p(x) in factored form.
241. Suppose g(x) = 2x
3 – 6x
2 – 2x 6 has zeros ±1 and 3. Express g(x) in
factored form.
242. Fill in the blank to make a true statement.
(A) The fundamental theorem of algebra states that, over the complex
numbers, every polynomial equation of degree n ≥ 1 has at least
_____________ root(s).
(B) If a root of a polynomial equation has multiplicity k, then that root will
occur exactly _____________ times in the list of all roots.
(C) The zeros of a polynomial p are the _____________ of the equation
p(x) = 0.
(D) The fundamental theorem of algebra guarantees that every polynomial
of degree n ≥ 1 has exactly _____________ zeros, some of which
might repeat.
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(E) If p(x) is a polynomial equation with real coefficients and x yi is a
root of p(x) = 0, then its complex conjugate _____________ is also a
root of p(x) = 0.
For questions 243 and 244, (a) factor p(x) completely, and (b) list all the zeros of
p. Refer to the following formulas, as needed.
acx
2 (ad bc)x bd = (ax b)(cx d) General trinomial
x
2 2xy y
2 = (x y)
2 Perfect square
x
2 – y
2 = (x y)(x – y) Difference of two squares
x
2 y
2 = (x yi)(x – yi) Sum of two squares
Note: Recall that i
2 = –1.
x
3 y
3 = (x y)(x
2 – xy y
2) Sum of two cubes
x
3 – y
3 = (x – y)(x
2 xy y
2) Difference of two cubes
Quadratic formula
243. p(x) = (x 3)(x
2 – 5)(x
2 – 49)(x
2 49)
244. p(x) = (3x
2 – x – 10)(x
6 – 64)
245. Find a polynomial equation p(x) with real coefficients and with the least
degree that has 3 and 2 – i as roots.
Descartes’ Rule of Signs and Rational Root
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