Download 2.13 Factors and Integral Roots – Day 2

2.13 Factors and Integral Roots – Day 2
1. Your teacher will assign your team one of the following polynomial functions. Use the list of
steps your team developed to factor the polynomial and find all of its roots. Then prepare a poster
in which you illustrate and justify each of your steps. Be sure to include the graph on your poster
and clearly explain the relationship between the solutions of the equation and the x-intercepts.
a.
Q1(x) = x3 + 3x2 + 1x − 5
b.
Q2(x) = 6x4 + 7x3 − 36x2 − 7x + 6
c.
Q3(x) = x4 + 2x3 + 10x2 + 18x + 9
d.
Q4 (x) = x4 − 8x3 + 18x2 − 16x + 5
e.
Q5(x) = x5 − 4x3 − x2 + 4
f.
Q6(x) = x3 + x2 − 7x − 7
2. LEARNING LOG/SUMMARY
Make additions and/or adjustments to your Summary entry from Lesson 2.12 to reflect what you
learned from the posters that you and your classmates presented for problem 1.
3. Use the procedures you developed to factor each of the following polynomial expressions. Each
final answer should be a linear factor times a quadratic factor. Look for patterns in the factors.
a. x3 − 1
b. x3 + 8
c. x3 − 27
d. x3 + 125
4. If you generalize the patterns for the factors in problem 3, you will discover a shortcut for
factoring cubic polynomials that are described as the sum or difference of two cubes. Write the
factors for each polynomial expression below.
a. x3 + a3
b.
x3 − b3
5. Are there similar patterns for x4 + a4 and x4 − b4? Explain.
6. BUILDING POLYNOMIALS
For each of the following descriptions of polynomial functions with integral coefficients, answer
each question below.
i.
What are the possible numbers of real zeros?
ii.
How many complex zeros are possible?
iii.
For each number of possible real zeros, give an example of a polynomial in factored
form.
a. A third-degree polynomial function.
b. A fourth-degree polynomial function.
c. A fifth-degree polynomial function.
Polynomial Theorems
Factor Theorem: If x = a is a zero of a polynomial function, then p(x) then (x − a) is a factor of
the polynomial, and if (x − a) is a factor of the polynomial, then x = a is a zero of the polynomial.
Example: If x = −3, x = 2, and x =
are zeros of the polynomial p(x) = 2x4 − x3 + 20x − 48.
According to the Factor Theorem, (x + 3),(x − 2),
the polynomial.
, and
, are factors of
Fundamental Theorem of Algebra: The Fundamental Theorem of Algebra states that every
one-variable polynomial has at least one complex root (remember that every real number is also a
complex number). This theorem can be used to show that every polynomial of
degree n has n complex roots. This also means that the polynomial has n linear factors, since for
every root a, (x – a) is a linear factor.
Note that the total of n roots might include roots that occur multiple times. For example, the third
degree polynomial p(x) = x3 − 3x2 + 4 has three roots—a root at x = –1 and a double root at
x = 2. The three linear factors are (x + 1), (x – 2), and (x – 2)
Integral Zero Theorem: For any polynomial with integral coefficients, if an integer is a zero of
the polynomial, it must be a factor of the constant term.
Example: Suppose the integers a, b, c, and d are zeros of a polynomial. Then, according to the
Factor Theorem, (x − a)(x − b)(x − c)(x − d) are factors of the polynomial.When you multiply
these factors together, the constant term will be abcd, so a, b, c, and d are factors of the constant
term.
Remainder Theorem: For any number c, when a polynomial p(x) is divided by (x – c), the
remainder is p(c). For example, if the polynomial p(x) = x4 −x3 + 20x –48 is divided by (x − 5),
the remainder is p(5) = 552.
Note that it follows from the Remainder Theorem that if p(c)= 0, then (x – c), is a factor. For
example, if p(x) = x3 −3x2 + 4, one solution to x3 −3x2 + 4 = 0 is x = –1. Since p(–1) = 0, then
(x + 1) is a factor.