Factoring Polynomials

Note. It is assumed that you know the properties of addition and multiplication
as explained in Section R.1. If you are not comfortable with
this, then please review (especially pages 10–13).

Note. We begin our adventure with a few definitions!

Definition. The integers are
Z = {. . . ,−3,−2,−1, 0, 1, 2, 3, . . .}.

The rational numbers (or “fractions”) are

definition. A polynomial in one variable x is an algebraic ex pression of
the form

where are constants called coefficients of the polynomial,
n ≥ 0 is an integer (i.e., n is some element of the set {0, 1, 2, . . .}),
and x is a variable. If , then an is the leading coefficient and n is
the degree of the polynomial.

Example. Some examples of polynomials are:

Example. Some functions which are not polynomials are:

Recall. A monomial is a polynomial with all but one coefficient 0:
for a ≠ 0 and k ≥ 0. A binomial is a polynomial with two nonzero
coefficients: where a ≠ 0, b ≠ 0, k > 0 > m ≥ 0, and k ≠ m
are integers. A trinomial is a polynomial with exactly three nonzero
coefficients: where a ≠ 0, b ≠ 0, c ≠ 0, k > m > n ≥ 0,
and k ≠ m ≠ n ≠ k are integers.

Note. We are mostly concerned with binomials of the form mx+b. these
are also called polynomials of degree 1. Our goals are to find when such
equations are 0 (Section 1.1) and to graph such equations (Section 4.1).

Note. We are also interested in trinomials of the form ax²+bx+c. This
is also called a polynomial of degree 2. Our goals are to find out when
such a polynomial is 0 (Section 1.2), to graph such polynomials (Sections
3.4 and 3.5), and applications of such polynomials (Section 4.1).

Definition. A binomial of degree one, ax+b, is a factor of a polynomial
p(x) if p(x) = (ax + b)r(x) for a polynomial r(x). If a, b, and the
coefficients of p(x) and r(x) are all integers, then ax + b is a factor over
the integers of p(x).

Example. Since (5x−2)(2x + 3) = 10x² +11x−6 (Right? Remember
FOIL!), then 5x − 2 and 2x + 3 are factors of 10x² + 11x − 6.

Definition. If a polynomial cannot be written as the product of two
other polynomials (excluding 1 and −1), then the polynomial is said to
be prime. When a polynomial is written as a product consisting only of
prime factors, it is said to be factored completely.

Example. We can completely factor 30x² + 33x − 18 as:

30x² + 33x − 18 = 3(10x² + 11x − 6) = 3(5x − 2)(2x + 3).

Notice that we hae used information we already knew (given in the example
above), we have used equal signs when things are equal, and we
have not written any unnecessary or incorrect symbols ! That is, we have
clearly and cleanly communicated our computations!!! Mathematics
consists of these two very important properties: accuracy
and clarity!!!

Example. Page 51 number 6.

Note. Based on FOIL, we have the fol lowing algebraic identities (i.e.,
equations which hold for every value of the variable x and for any values
of the coefficient a):

Example. Page 51 numbers 20, 26, 34 ,and 36.

Note. We will see in the not-to-distant future that we can factor any
second degree polynomial ax² + bx + c. For now, we concentrate on
second degree polynomials of the form x² + Bx + C.

Note. To factor x² + Bx + C, where B and C are integers, find integers
a and b whose product is C and whose sum is B. That is, find integers a
and b where B = a+b and C = ab. Then x² +Bx+C = (x+a)(x+b).

Example. Page 46 Example 9. Page 51 numbers 40 and 48.

Note. Until the very last section of this course (Section 4.7), we will only
study real numbers R (as opposed to complex numbers C). Whenever a
real number a is squared , the result is non negative : a² ≥ 0 for all a ∈ R.
Therefore we have:

Theorem. Any polynomial of the form x² + a², a real, is prime.

Note. One can show that every polynomial with real coefficients can be
factored into prime factors which consist of prime first degree polynomials
and prime second degree polynomials with real coefficients (Theorem on
page 37, Section 4.6).

Note. We can also factor polynomials “ by grouping .” In this method we
recognize common factors and take advantage of the distributive law of
multiplication over division .

Example. Page 51 numbers 52 and 56.

Note. Now we tackle the task of factoring the second degree polynomial
Ax² + Bx + C where A ≠1. We follow these steps :

1. Find the value of AC.

2. Find integers with product AC that add up to B. That is, find integers
ab such that AC = ab and B = a + b.

3. Write Ax² + Bx + C = Ax² + ax + bx + C.

4. Factor by grouping.

Example. Page 51 numbers 62, 68, 100, and 124.