Algebra – Fundamentals
Fundamentals in Algebra
Key ideas
- Real numbers and algebraic expressions follow strict operational rules like the commutative and associative properties.
- Exponent and radical rules allow you to simplify complex expressions systematically.
- Special products provide reliable shortcuts for expanding binomials and trinomials without manual distribution.
- The remainder theorem offers a fast way to find the remainder of a polynomial division simply by evaluating a function.
Properties of numbers and equality
Algebra relies on foundational properties that dictate how numbers and variables interact. The number 0 acts as the additive identity, while 1 is the multiplicative identity.
| Property | Addition | Multiplication |
|---|---|---|
| Closure | \( a + b \) is a real number | \( a \cdot b \) is a real number |
| Commutative | \( a + b = b + a \) | \( a \cdot b = b \cdot a \) |
| Associative | \( (a + b) + c = a + (b + c) \) | \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) |
| Identity | \( a + 0 = a \) | \( a \cdot 1 = a \) |
| Inverse | \( a + (-a) = 0 \) | \( a \cdot \left(\dfrac{1}{a}\right) = 1 \) |
| Distributive | \( (a + b)c = ac + bc \) and \( a(b + c) = ab + ac \) | |
Equality between mathematical expressions is governed by specific logical rules.
- Reflexive property states that \( a = a \).
- Symmetric property states that if \( a = b \), then \( b = a \).
- Transitive property states that if \( a = b \) and \( b = c \), then \( a = c \).
- Substitution property allows \( a \) to be replaced by \( b \) in any expression if \( a = b \).
- Cancellation property allows you to simplify equations, meaning if \( a + c = b + c \), then \( a = b \).
Zero has unique mathematical behaviors. Adding or subtracting zero leaves a number unchanged. Multiplying any number by zero results in zero. Dividing zero by any non-zero number yields zero, but dividing a number by zero is strictly undefined.
Zero-factor property states that if the product of two numbers is zero (\( ab = 0 \)), then either \( a = 0 \) or \( b = 0 \).
Exponents and radicals
The exponential notation \( a^n \) represents the base \( a \) multiplied by itself \( n \) times. A radical expression \( \sqrt[n]{a} \) consists of an index \( n \) and a radicand \( a \).
Example. Simplify the expression by applying the power of a quotient property to \( \left(\dfrac{5}{b}\right)^2 \).
Solution. Distribute the exponent to both the numerator and the denominator.
Substitute the given variables into the property.
Compute the numerical base raised to the power.
Answer: \( \dfrac{25}{b^2} \)
Example. Evaluate the product of two radicals: \( \sqrt[3]{2} \cdot \sqrt[3]{500} \).
Solution. Apply the property for multiplying radicals with identical indices.
Substitute the specific numbers into the formula.
Extract the exact cube root of the resulting radicand.
Answer: \( 10 \)
Surd is an irrational number expressed strictly as a radical.
Pure surd contains no rational numbers outside the radical symbol.
Mixed surd contains at least one rational factor attached to the radical.
Binomial surd is a two-term algebraic expression where at least one term is a surd.
Trinomial surd is a three-term expression where at least two terms are surds that cannot be combined.
Proportions and theorems
Proportion is an equation stating that two ratios are entirely equal.
Antecedent is the first term or numerator in a mathematical ratio.
Consequent is the second term or denominator in a mathematical ratio.
Extremes are the outer terms in a proportion, such as \( a \) and \( d \) in \( a:x = y:d \).
Means are the inner terms in a proportion, such as \( x \) and \( y \).
Mean proportional occurs when the two inner terms of a proportion are identical.
Remainder theorem states that if a polynomial is divided by a linear expression \( (x - k) \), the resulting remainder equals the numerical value of the polynomial evaluated at \( k \).
Factor theorem states that if substituting \( k \) into a polynomial yields a remainder of exactly zero, then \( (x - k) \) is a definitive factor of that polynomial.
Multiples and factors
Least common denominator (LCD) is the smallest multiple shared by all prime factors in the denominators of a given set of fractions.
Least common multiple (LCM) is the smallest possible integer that is a multiple of every given number in a set.
Greatest common factor (GCF) is the largest integer value that divides all the given numbers evenly.
Example. Determine the least common multiple (LCM) of 12 and 16.
Solution. Break down each number into its prime factorization.
Identify and multiply the highest power of each prime factor present.
Compute the final numerical product.
Answer: \( 48 \)
Example. Find the greatest common factor (GCF) of 48 and 72.
Solution. Write out the complete prime factorization for both numbers.
Select the lowest power of each shared prime factor.
Multiply these common factors together to get the result.
Answer: \( 24 \)
Additional facts. Both the remainder theorem and the factor theorem were introduced by the French mathematician Etienne Bezout. Furthermore, the familiar symbol for equality consisting of two long parallel lines (=) was first introduced by Robert Recorde in 1557.
Formulas
- Product of powers: \( a^m \cdot a^n = a^{m+n} \)
- Quotient of powers: \( \dfrac{a^m}{a^n} = a^{m-n} \)
- Power of a power: \( (a^m)^n = a^{mn} \)
- Negative exponent: \( a^{-m} = \dfrac{1}{a^m} \)
- Fractional exponent: \( a^{m/n} = \sqrt[n]{a^m} \)
- Difference of two squares: \( (x+y)(x-y) = x^2 - y^2 \)
- Square of a binomial: \( (x \pm y)^2 = x^2 \pm 2xy + y^2 \)
- Cube of a binomial: \( (x \pm y)^3 = x^3 \pm 3x^2y + 3xy^2 \pm y^3 \)
- Difference of two cubes: \( x^3 - y^3 = (x-y)(x^2 + xy + y^2) \)
- Sum of two cubes: \( x^3 + y^3 = (x+y)(x^2 - xy + y^2) \)
- Square of a trinomial: \( (x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz \)
Quick check
Why is division by zero strictly undefined in algebraic operations?
Show answer
Because there is no real number that, when multiplied by zero, can produce a non-zero numerator. The inverse operation simply breaks down.
What is the fundamental difference between a prime factor and a greatest common factor?
Show answer
A prime factor is an indivisible building block of a single number, while the greatest common factor is the largest shared value that divides two or more separate numbers evenly.