Algebra

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.

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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).

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