The Rules of Algebra

Robert S. Wilson Ph.D.

 

            Algebra is arithmetic with numbers and letters. Arithmetic consists of the following operations

 

                        Addition                      Subtraction

                        Multiplication              Division

                        Powers                        Roots and Logarithms

 

            Addition is repeated counting. Subtraction reverses addition. Multiplication is repeated addition. Division reverses multiplication. Powers or exponents are repeated multiplication. Roots and logarithms reverse powers.

 

            To apply an operation to two letters, write the letters down with the operation between them,

 

                        a + b                            a – b

                        ab                                a/b

                        ab                                    or  logba

 

            The answer to an arithmetic problem with letters is an expression. We can also do arithmetic with expressions. Copy down the expressions in parentheses and put the operation between the sets of parentheses.

 

                        (2x) + (a + b)              (2x) - (a + b)

                        (2x)(a + b)                   (2x)/(a + b)   or   

                        (a + b)2x                             or    log2x(a + b)

 

            A fraction is the answer to a division problem. If you use the vertical format, the fraction bar acts like parentheses.

 

The Order of Operations

            Combine what’s inside parentheses.

            Exponents and roots

            Multiplications and divisions

            Additions and subtractions.

 

Rules for Parentheses

            a + b = b + a                           ab = ba                        Commutative rules

                        a + (b + c) = (a + b) + c          a(bc) = (ab)c               Associative rules

                        0 + a = a                                 1a = a                          Identity rules

                        -a + a = 0                                a  = 1                       Inverse rules    (a ≠ 0)

                                    a( + c) = ab + ac                               Distributive rule

 

Rules for Exponents

 

            Definition       an = a a . . . a

                                           n factors               (n = 1, 2, 3, . . . )

 

anam = a(n+ m)                           (ab)n = anbn

(an)m = anm                             

                              

a0 = 1

 

Definitions

 

Terms are added and subtracted

Factors are multiplied and divided

 

There are two kids of factors, number factors and letter factors. There need only be one numerical factor in a term, because the commutative and associative rules enable you to move all of your numerical factors together and multiply them up to get a single numerical factor, which the commutative rule says you can write at the left hand side of the term. This numerical factor is called the coefficient. The letter factors are called variables. If your term contains several factors of the same variable, you can tell your reader how many factors of that variable there are by using a power or an exponent. The number of variable factors in a term is called the degree of the term.

 

An expression which is made up of only addition, subtraction, and multiplication is called a polynomial. The coefficients in a polynomial can be fractions, but there are no variables in denominators. The degree of a polynomial is the degree of the highest degree term. Polynomials of degree one are called linear. Polynomials of degree two are called quadratic. Polynomials of degree three are called cubic. Polynomials of degree four are called quartic. Polynomials of degree five are called quintic. A polynomial with one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial.

 

Like terms are terms with exactly the same variable factors. The distributive property enables you to combine like terms. If  the two terms have exactly the same variable factors, you can factor them all out leaving nothing but numbers inside parentheses, and whenever you have nothing but numbers inside parentheses, you can combine them into a single numbers. To combine like terms, add or subtract the coefficients. The common variable factors give us the variables in the answer.

 

The distributive property also allows us to remove parentheses. To remove parentheses, multiply the factor outside the parentheses by all the terms inside the parentheses, and add the products.

 

Arithmetic with Polynomials

 

To add or subtract or multiply polynomials, remove parentheses and combine like terms. For multiplying, this amounts to multiplying each term in one polynomial by each term in the other. To multiply terms, multiply the coefficients and add the exponents on each variable. The number of terms in the product will be equal to the product of the number of terms. Of course, there may well be like terms which you will need to combine.

 

Long Division of Polynomials

 

Definitions

 

The polynomial you are dividing by is called the divisor.

The polynomial you are dividing it into is called the dividend.

The answer is called the quotient.

The difference between the quotient times the divisor and the dividend is called the remainder.

 

1.              Divide the first term in the dividend into the first term in the divisor. This gives you the first term in the quotient.

2.              Multiply this term in the quotient by the divisor and subtract this product from the dividend.

3.              Repeat the process with the remainder until you have a remainder whose degree is smaller than the degree of the divisor.

 

Fractions

 

Definition:  A fraction (or rational expression) is the answer to a division problem of polynomials.

 

The Fundamental Fact of Fractions

 

If you multiply (or divide) the top and bottom of a fraction by the same thing, you get a different name for the same number.

 

Reducing or Simplifying Fractions

 

            Factor the top and bottom until you get factors that cannot be factored further. If you find the same factor on both the top and bottom, you can cancel them. If after factoring the top and bottom as much as possible, if there are no common factors in the top and bottom, the fraction is reduced to lowest terms.

 

Adding (Subtracting) Fractions

 

            If you have common denominators, add (subtract) the numerators.

 

            If not, find common denominators.

 

            To find common denominators, factor all the denominators and fill in the missing factors. You can multiply the bottom by whatever you want so long as you multiply the top by the same thing.

 

Multiplying Fractions

 

            Multiply the tops and multiply the bottoms.

            You can cancel either before or after you multiply.

 

Dividing Fractions

 

            Invert the divisor and multiply.

 

Equations

 

            Def’n:  An equation consists of an equals sign with and expression on either side.

 

            Def’n:  A solution to an equation is something you can substitute in for a variable in an equation, which would make the same thing come out on both sides.

 

            Def”n:  Two equations are equivalent if they have the same solution(s)

 

The Fundamental Technique for Equations

 

            You can make any change you want on one side of an equation so long as you make the same change on the other side.

 

            One of the most common techniques is to get rid of a term on one side by subtracting it from both sides. When you get rid of a term on one side, it pops up on the other side with its sign changed. Moving a term from one side to the other and changing its sign is called transposing the term.

 

            If you move a factor from one side to the other, move it across the fraction bar.

 

Steps in solving first degree equations

 

1.              Clear Denominators: Multiply both sides by a common denominator.

2.              Simplify: Remove parentheses and combine like terms.

3.              Transpose known terms to one side and unknown terms to the other.

4.              Combine.

5.              Divide both sides by the coefficient of the unknown.

 

Steps in solving quadratic equations by factoring.

 

  1. Clear Denominators: Multiply both sides by a common denominator.
  2. Simplify: Remove parentheses and combine like terms.
  3. Transpose all terms to one side leaving a  0  on the other.
  4. Combine.
  5. Factor.
  6. Set the factors = 0.
  7. Solve the two resulting first degree equations.
  8. Check.

 

Steps in solving quadratic equations by completing the square.

 

  1. Clear Denominators: Multiply both sides by a common denominator.
  2. Simplify: Remove parentheses and combine like terms.
  3. Transpose known terms to one side and unknown terms to the other
  4. Combine
  5. Divide both sides by the coefficient of the square term.
  6. Add the square of half the coefficient of the first degree term to both sides.
  7. Simplify to get a perfect square on one side and a number on the other.
  8. Take square roots of both sides.
  9. Transpose the number with the unknown to the other side.
  10. Check.

 

Steps in solving quadratic equations using the quadratic formula.

 

1.              Clear denominators: Multiply both sides by a common denominator.

2.              Simplify: Remove parentheses and combine like terms.

3.              Transpose all terms to one side leaving a 0 on the other.

4.              Combine.

5.              Substitute the coefficients into the quadratic formula.

 

6.              Check.

 

Polynomial equations of degree higher than two are beyond the scope of this discussion. There is a cubic and a quartic formula, involving radicals, to get rid of the powers, but beyond that it can be proven that solution by radicals will not always work. One reason the cubic and quartic formulas are not often studied is because they are so complicated as to be computationally useless.

 

Steps in solving rational equations.

 

 

After clearing denominators, you will have a polynomial equation. If it is first or second degree, the steps above wiill suffice.

 

Logarithms.

 

If   bx = a,   then   x = logb a

 

 

logb bx = x

 

 

logb xy = logb x + logb y

 

logb xr = rlogb x

 

 

 

 

 

change of base formula