### OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA

**Important Definitions**

Factors; coefficient; exponent; power; base; term; algebraic sum;
similar terms; degree; homogeneous expression; linear equation; root of
an equation; root of an expression; identity; conditional equation;
prime quantity; highest common factor (H. C. F.); lowest common multiple
(L. C. M.); involution; evolution; imaginary number; real number;
rational; similar radicals; binomial surd; pure quadratic equation;
affected quadratic equation; equation in the quadratic form;
simultaneous linear equations; simultaneous quadratic equations;
discriminant; symmetrical expression; ratio; proportion; fourth
proportional; third proportional; mean proportional; arithmetic
progression; geometric progression;

**Special Rules for Multiplication and Division**

1. Square of the sum of two quantities.

2. Square of the difference of two quantities.

3. Product of the sum and difference of two quantities.

4. Product of two binomials having a common term.

5. Product of two binomials whose corresponding terms are similar.

6. Square of a polynomial.

7. Sum of two cubes.

8. Difference of two cubes.

9. Sum or difference of two like powers.

**Cases in Factoring**

1. Common monomial factor.

2. Trinomial that is a perfect square.

3. The difference of two squares.

5. Trinomial of the form

6. Sum or difference of

8. Factor Theorem.

(a) Two terms.

(b) Four terms.

(c) Six terms.

(d) Incomplete square.

4. Trinomial of the form 6. Sum or difference of

- two cubes. See "Special Rules," 7 and 8.
- two like powers. See "Special Rules," 9.

**H. C. F. and L. C. M.**

H. C. F.

L. C. M.

**Fractions**

Reduction to lowest terms.

Reduction of a mixed number to an improper fraction.

Reduction of an improper fraction to a mixed number.

Addition and subtraction of fractions.

Multiplication and division of fractions.

Law of signs in division, changing signs of factors, etc.

Complex fractions.

Reduction of a mixed number to an improper fraction.

Reduction of an improper fraction to a mixed number.

Addition and subtraction of fractions.

Multiplication and division of fractions.

Law of signs in division, changing signs of factors, etc.

Complex fractions.

**Simultaneous Equations**

Solved by

- addition or subtraction.
- substitution.
- comparison.

**Involution**

Law of signs.

Binomial theorem laws.

Expansion of

Binomial theorem laws.

Expansion of

- monomials and fractions.
- binomials.
- trinomials.

**Evolution**

Law of signs.

Evolution of monomials and fractions.

Square root of algebraic expressions.

Square root of arithmetical numbers.

Optional

Evolution of monomials and fractions.

Square root of algebraic expressions.

Square root of arithmetical numbers.

Optional

- Cube root of algebraic expressions.
- Cube root of arithmetical numbers.

**Theory of Exponents**

Proofs:

Meaning of

Meaning of

- fractional exponent.
- zero exponent.
- negative exponent.

- To multiply quantities having the same base, add exponents.
- To divide quantities having the same base, subtract exponents.
- To raise to a power, multiply exponents.
- To extract a root, divide the exponent of the power by the index of the root.

**Radicals**

Radical in its simplest form.

Transformation of radicals

Multiplication and division of radicals

Rationalization

Radical equations.

Transformation of radicals

- Fraction under the radical sign.
- Reduction to an entire surd.
- Changing to surds of different order.
- Reduction to simplest form.

Multiplication and division of radicals

- Monomial denominator.
- Binomial denominator.
- Trinomial denominator.

Radical equations.

*Always*check results to avoid extraneous roots.**Quadratic Equations**

Pure.

Affected.

Methods of solvingAffected.

- Completing the square.
- Formula. Developed from
- Factoring.

Properties of quadratics

- Then
- Discriminant, and its discussion.
- Nature or character of the roots.

**Simultaneous Quadratics**

Case I.

One equation linear.

The other quadratic.

Case II.The other quadratic.

Both equations homogeneous and of the second degree.

Case III.
Any two of the quantities etc., given.

Case IV.
Both equations symmetrical or symmetrical except for sign. Usually one equation of high degree, the other of the first degree.

Case V. Special Devices
I. Solve for a compound unknown, like etc., first.

II. Divide the equations, member by member.

III. Eliminate the quadratic terms.

**Ratio and Proportion**

Proportionals

- mean,
- third,
- fourth.

- Product of means equals product of extremes.
- If the product of two numbers equals the product of two other numbers, either pair, etc.
- Alternation.
- Inversion.
- Composition.
- Division.
- Composition and division.
- In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent, etc.

**Progressions**

Development of formulas.

Insertion of means

- Arithmetical.
- Geometrical.

**Binomial Theorem**

Review of binomial theorem laws. See Involution.

Expansion of

Finding any term by

Expansion of

Finding any term by

- key number method.
- or term method.

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