III.

Putting Polynomial Equations Into Solvable Forms

Solving a polynomial equation usually requires altering its form. The most common tools used to manipulate equations into solvable form are the laws of arithmetic, factoring, and the finding of least common multiples.

A.

Laws of Polynomial Arithmetic

In manipulating polynomials, the usual laws of the arithmetic of numbers hold. This section lays out those laws. Normal arithmetic is limited to the set of rational numbers (positive and negative whole numbers and infinitely repeating decimals). Algebra and geometry can include irrational numbers (infinite decimals that do not repeat), such as pi (symbol p) and the square root of 2 (abbreviated √2). The set of all rational and irrational numbers taken together constitutes the set of real numbers. See also Number (mathematics).

A.1.

Laws of Addition

1. The sum of any two real numbers a and b is also a real number, denoted a + b. The real numbers are closed under the operations of addition, subtraction, multiplication, division, and the extraction of roots; this means that applying any of these operations to real numbers yields a quantity that also is a real number.

2. No matter how terms are grouped in carrying out additions, the sum will always be the same: (a + b) + c = a + (b + c). This is called the associative law of addition.

3. Given any real number a, there is a real number zero (0) called the additive identity, such that a + 0 = 0 + a = a.

4. Given any real number a, there is a number (-a), called the additive inverse of a, such that (a) + (-a) = 0.

5. No matter in what order addition is carried out, the sum will always be the same: a + b = b + a. This is called the commutative law of addition.

Any set of numbers obeying laws 1 through 4 is said to form a group. If the set also obeys law 5, it is said to be an Abelian, or commutative, group. Integers and real numbers are both Abelian groups. Since subtraction can be treated as the addition of negative numbers (3 – 4 is the same as 3 + -4), these laws also apply to subtraction.

A.2.

Laws of Multiplication

Laws similar to those for addition also apply to multiplication. Since powers are a special case of multiplication, these laws cover powers as well.

1. The product of any two real numbers a and b is also a real number, denoted a·b or ab.

2. No matter how terms are grouped in carrying out multiplications, the product will always be the same: (ab)c = a(bc). This is called the associative law of multiplication.

3. Given any real number a, there is a number one (1) called the multiplicative identity, such that a(1) = 1(a) = a.

4. Given any nonzero real number a, there is a number (a^-1), or (1/a), called the multiplicative inverse, such that a(a^-1) = (a^-1)a = 1.

5. No matter in what order multiplication is carried out, the product will always be the same: ab = ba. This is called the commutative law of multiplication.

Any set of elements obeying these five laws is said to be an Abelian, or commutative, group under multiplication. The set of all real numbers, excluding zero (because division by zero is impossible), forms such a commutative group under multiplication. Law number 4 allows the laws of multiplication to be extended to division, since dividing by a number is the same as multiplying by its inverse.

A.3.

Distributive Laws

Another important property of the set of real numbers links addition and multiplication in two distributive laws as follows:

1. a(b + c) = ab + ac

2. (b + c)a = ba + ca

Any set of elements with an equality relation and for which two operations (such as addition and multiplication) are defined, and which obeys all the laws for addition, the laws for multiplication, and the distributive laws, constitutes a field.

A.4.

Exponent Laws

The laws governing exponents are extensions of the addition, multiplication, and distributive laws. Since a² = a · a and a³ = a · a · a, for example, a² · a³ = a · a · a · a · a = a²⁺³ = a⁵. In general, therefore, for real numbers:

1. a^m·aⁿ = a^m+n

2. (a^m)ⁿ = a^m·n

3. a^m·b^m = (a·b)^m

B.

Multiplying Polynomials

To multiply polynomials, each term of each polynomial is multiplied in turn by all of the terms in all of the other polynomials. The results are then added together. The following is a simple example of the product of a binomial and a monomial: (x + 2)(4x²) = x(4x²) + 2(4x²) = 4x³ + 8x²

Each term of the one polynomial, (x + 2), is multiplied by the single term of the monomial, 4x². This same principle is directly extended to polynomials of any number of terms. For example, the product of a binomial and a trinomial is carried out as follows:

(5x + 2)(3x³ + x² -4x) =

[5x(3x³ + x² -4x)] + [2(3x³ + x² -4x)] =

[5x(3x³) + 5x(x²) + 5x(-4x)] + [2(3x³) + 2(x²) + 2(-4x)] =

[15x⁴ + 5x³ – 20x²] + [6x³ + 2x² -8x] =

After such operations have been performed, all terms of the same degree should be combined whenever possible to simplify the entire expression:

15x⁴ + 11x³ - 18x² - 8x

C.

Factoring Polynomials

Given a complicated algebraic expression, it is often useful to factor it into the product of simpler terms. For example, 2x² + 4xy can be factored as 2x(x + 2y) because (x + 2y) multiplied by 2x is equal to 2x(x) + 2x(2y), which simplifies to 2x² + 4xy. Determining the factors of a given polynomial may be a simple matter of inspection or may require trial and error. Not all polynomials, however, can be factored using real-number coefficients, and these are called prime polynomials. Some common factorizations are given in the following examples.

1. Trinomials of the general form x² + (a + b)x + ab can be factored (x + a)(x + b). If a = 5 and b = 2, for example, x² + 7x + 10 can be factored (x + 5)(x + 2). The problem can be worked backwards by multiplying out the factors to prove this factoring is correct:

(x + 5)(x + 2) =

x(x + 2) + 5(x + 2) =

x² + 2x + 5x + 10 =

x² + 7x + 10

2. Trinomials of the general form a²x² + (2ab)(xy) + b²y² can be factored (ax + by)(ax + by) or simply (ax + by)². In the simple example where a and b are both equal to 1, x² + 2xy + y² can be factored (x + y)²:

(x + y)² =

(x + y)(x + y) =

x² + xy + xy + y² =

x² +2xy + y²

Similarly, when a = 1 and b = -1, x² – 2xy +y² = (x – y)². In a more complex example, where a = 5 and b = 2, 25x² + 20xy + 4y² can be factored (5x + 2y)²:

(5x + 2y)² =

(5x + 2y) (5x + 2y) =

25x² + 10xy + 10xy + 4y² =

25x² + 20xy + 2y²

3. The difference of squares of the form a²x² – b²y² may be factored (ax + by)(ax – by). In the simplest case, when a and b equal 1, x² - y² = (x + y)(x – y) because the terms of lower degree add up to zero and thus cancel each other out:

(x + y)(x – y) =

x² – xy + xy – y² =

x² - y²

In the more complex case where a = 5 and b = 4, 25x²-16y² = (5x + 4y)(5x – 4y):

(5x + 4y)(5x – 4y) =

25x² – 20xy + 20xy – 16y² =

25x² – 16y²

4. The sums and differences of cubes can also be factored according to the general formulas a³x³ + b³y³ = (ax + by)(a²x² – axby + b²y²) and a³x³ - b³y³ = (ax - by)(a²x² – axby + b²y²). When a and b are both equal to 1, x³ + y³ = (x + y)(x² - xy + y²):

(x + y)(x² - xy + y²) =

x³ - x²y + xy² + x²y - xy² + y³ =

x³ + y³

and x³ - y³ = (x - y)(x² + xy + y²).

Grouping may often be useful in factoring; terms that are similar are grouped wherever possible, as in the following example: 10x³ + 16x²y + 18xy² -2x³ + 8x²y = 8x³ + 24x²y + 18xy² = 2x(4x² +12xy + 9y²) = 2x(2x+3y)².

D.

Identifying Highest Common Factors

Given a polynomial, it is frequently important to isolate the greatest common factor from each term of the polynomial. For example, in the binomial 6x² + 12x, the number 6 is a factor of both terms, as is x. After factoring, 6x(x + 2) is obtained, and 6x is the greatest common factor for all terms of the original binomial. Similarly, for the trinomial 6a²x³ + 9abx + 15cx², the number 3 is the largest numerical factor common to 6, 9, and 15, and x is the largest variable factor common to all three terms. Thus, the greatest common factor of the trinomial is 3x and the trinomial can be factored 3x(2a²x² + 3ab + 5cx).

E.

Identifying Least Common Multiples

Finding least common multiples is useful in combining algebraic fractions. The least common multiple (LCM) of a set of numbers is the smallest number into which each number in the set will divide evenly. The LCM of 2, 3, 4, and 6, for example, is 12.

The LCM can be calculated by factoring numbers into their prime components. The LCM is the product of the highest power of each prime factor of the given numbers. For example, to find the LCM for the three numbers 27, 63, and 75, each number is first factored: 27 = 3³, 63 = 3²·7, and 75 = 3·5². The prime factors of these three numbers are 3, 5, and 7, and the highest powers of those three factors are 3³, 5², and 7. The LCM, therefore, is 3³·7·5² = 4,725; 4,725 is the smallest number into which 27, 63, and 75 will all divide evenly.

Given several algebraic expressions, the least common multiple is the expression of lowest degree and least coefficient that can be divided evenly by each of the expressions. To find a common multiple of the terms 2x²y, 15x²y², and 6ay³, all three expressions could simply be multiplied together: (2x²y)(15x²y²)(6ay³) = 180ax⁴y⁶. However, 180ax⁴y⁶ is not the least common multiple. To determine which is the least, each of the terms is reduced to its prime factors. For the numerical coefficients 2, 15, and 6, the prime factors are 2, 3·5, and 2·3, respectively; the least common multiple for the numerical coefficients is therefore 2·3·5, or 30. Similarly, because the constant a appears only once, it too must be a factor. Of the variables, x² and y³ are required because they are the highest powers of the two variables that appear in any of the expressions. The LCM of the three terms, therefore, is 30ax²y³. Each term will evenly divide this expression: