Algebraic Operations

If you build a square fenced garden, what is its perimeter? If each side is $2\text{ m}$, perimeter is $4 \times 2 = 8\text{ m}$. If each side is $6\text{ m}$, perimeter is $4 \times 6 = 24\text{ m}$. The number 4 is fixed (constant), while the side length changes. We represent this variable side length by the letter $x$, creating the algebraic formula $\mathbf{4x}$.

• Variable: A symbol (usually letters $x, y, z, a, b$) that represents an unknown or changing quantity.
• Constant: A symbol with a fixed, unvarying numerical value ($7, -15, \frac{3}{4}$).
• Algebraic Term: A combination of constants and variables linked strictly by multiplication or division. Terms are separated from one another by $+$ or $-$ signs. In $5x^2 - 4xy + 9$, the terms are $5x^2, -4xy, 9$.
• Coefficient: The multiplier associated with a variable factor:
  • Numerical Coefficient: The constant multiplier including its sign. In $-7x^2y$, the numerical coefficient is $\mathbf{-7}$.
  • Literal Coefficient: In $-7x^2y$, the coefficient of $y$ is $-7x^2$, and the coefficient of $x^2$ is $-7y$.
• Polynomial Classification: Monomial (1 term: $4x$), Binomial (2 terms: $2x - 3$), Trinomial (3 terms: $x^2 + 4x + 4$), Polynomial (any expression with 1 or more terms).

3 dollars plus 2 dollars equals 5 dollars. But 3 dollars plus 2 euros cannot merge into "5 dollar-euros"—they are different currencies. In algebra, only terms with the exact same variable powers are "the same currency" (Like Terms). Unlike terms must remain separate.

• Like Terms: Terms whose variable factors and corresponding powers are completely identical (e.g. $7x^2y$ and $-3x^2y$). Only numerical coefficients differ.
• Unlike Terms: Terms having different variables or different powers of the same variable (e.g. $4x$ and $4x^2$, or $5xy$ and $5x$).
• Addition Rule: Combine like terms by adding their numerical coefficients; variable factors remain unchanged:
  $$ax^n + bx^n = (a + b)x^n$$
• Subtraction Rule (Sign Reversal): When subtracting an expression, distribute the minus sign to reverse every sign inside the subtrahend:
  $$-(a - b + c) = -a + b - c$$

A rectangle with length $(x + 4)$ and breadth $(x + 3)$ has an area equal to $(x + 4)(x + 3)$. Splitting the rectangle into 4 smaller sub-rectangles gives areas $x^2$, $3x$, $4x$, and $12$. Total area $= x^2 + 7x + 12$. Multiplying algebraic expressions is geometric distribution!

• Monomial by Monomial:
  1) Multiply signs: $(+) \times (+) = +$, $(-) \times (-) = +$, $(+) \times (-) = -$
  2) Multiply numerical coefficients
  3) Add powers of identical bases ($x^m \times x^n = x^{m+n}$)
  $$\mathbf{(ax^m) \times (bx^n) = (ab)x^{m+n}}$$
• Monomial by Polynomial (Distributive Law):
  $$\mathbf{a(b + c - d) = ab + ac - ad}$$
• Binomial by Binomial:
  $$\mathbf{(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd}$$

Division is inverse multiplication: $\frac{20x^4}{5x^2} = ?$ asks "What multiplied by $5x^2$ yields $20x^4$?". Since $4 \times 5 = 20$ and $x^2 \times x^2 = x^4$, the quotient is $4x^2$. Coefficients divide while exponents subtract.

• Monomial by Monomial:
  Divide numerical coefficients, and apply Quotient Law of Indices ($x^m \div x^n = x^{m-n}$):
  $$\mathbf{\frac{ax^m}{bx^n} = \left(\frac{a}{b}\right)x^{m-n}} \quad (b \neq 0, x \neq 0)$$
• Polynomial by Monomial (Term-by-Term Distribution):
  Divide every individual term in the numerator by the denominator:
  $$\mathbf{\frac{A + B - C}{M} = \frac{A}{M} + \frac{B}{M} - \frac{C}{M}}$$
• Division Verification Algorithm:
  $$\mathbf{\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}}$$

Algebraic formulas are general templates. To extract concrete answers for a specific case, you substitute numerical values into the variables. In physics, the kinetic energy equation $E_k = \frac{1}{2}mv^2$ yields the exact impact energy once mass ($m$) and velocity ($v$) are plugged in.

• Bracket Substitution Rule: Always enclose substituted numbers in parentheses $( )$, especially when negative numbers or exponents are present.
• Evaluation Precedence:
  1) Resolve powers/exponents first ($(-3)^2 = +9$)
  2) Perform multiplications and divisions
  3) Perform additions and subtractions