Algebraic Expressions, Monomial Rules, and Binomial Operations Review

Fundamentals of Algebraic Expressions and Tiles

Visual representations in algebra utilize geometric algebra tiles to model variable terms and numerical values. The standard tile set includes the unit tile representing the constant value $1$, the $x$ tile representing linear terms, the $x^2$ tile representing quadratic terms, and the cube tile representing $x^3$ or cubic terms. A foundational command in all algebraic manipulations is to always remember and track the sign of each term. A monomial is defined as an algebraic expression consisting of a single term, which can be a single number, a single variable, or a product of numbers and variables.

Operations on Signed Numbers and Like Terms

When performing addition and subtraction on algebraic terms, calculations rely on integer operational rules. For algebraic subtraction, the Keep, Change, Change (KCC) rule is applied: keep the sign of the first term, change the subtraction operator to addition, and change the sign of the second term.

When terms share the same sign during addition, add their values and keep the common sign. The sign rules for addition dictate that combining two positive values yields a positive result ($+ + = +$), while combining two negative values yields a negative result ($- - = -$). When terms have different signs, subtract the smaller magnitude from the larger magnitude and take the sign of the term with the larger absolute value.

For example, combining like monomial terms with different signs such as $5x^2 + (-3x^2)$ results in $2x^2$. Applying subtraction rules to terms such as $-5a$ and $2a$ using KCC transforms the expression to $-5a + (-2a)$, which yields $-7a$.

Exponent Rules for Monomials

The Product Rule states that when multiplying algebraic terms with exponent powers, add the exponents together. The product rule only works when the bases are identical, such as both bases being $x$ or both bases being $a$.

This rule is demonstrated by expanding the product $x^3 \times x^4$:

$x^3 \times x^4 = (x \cdot x \cdot x) \times (x \cdot x \cdot x \cdot x)$

$x^3 \times x^4 = x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x = x^7$

When the bases are the same, adding the exponents directly ($3 + 4 = 7$) yields the result $x^7$. The exponent notation $x^2$ represents $x$ squared or $x$ raised to the power of $2$.

The Quotient Rule states that when dividing algebraic terms, subtract the exponent of the denominator from the exponent of the numerator. Like the product rule, the quotient rule only works when the bases are identical.

Multiplication and Division of Monomials

Multiplication and division of monomials require evaluating the numerical coefficients and the variable bases independently. First, apply sign rules for multiplication and division to the numerical coefficients: terms with unlike signs yield a negative result, while terms with the same sign yield a positive result.

The complete sign multiplication rules are:

$\text{positive} \times \text{positive} = \text{positive}$

$\text{negative} \times \text{negative} = \text{positive}$

$\text{positive} \times \text{negative} = \text{negative}$

$\text{negative} \times \text{positive} = \text{negative}$

To multiply monomials with multiple variable bases, multiply the numerical coefficients together and apply the product rule to each corresponding variable base by adding their exponents. For example:

$(9x^3 y^4)(4x^2 y^3) = 36x^5 y^7$

For division of monomials, divide the numerical coefficients and apply the quotient rule to variable exponents by subtracting the denominator exponent from the numerator exponent. For example:

$\frac{36x^3 y^{12}}{-4x^5 y^3} = -9x^{-2} y^9 = \frac{-9y^9}{x^2}$

Addition and Subtraction of Trinomials

Adding and subtracting trinomials involves grouping and combining like variable terms and constant terms across all given polynomial expressions.

Consider the vertical column addition of three stacked trinomials:

$-m - 9n + 16$

$-5m - 18n + 11$

$-3m + 13n - 21$

Combining the $m$ variable terms yields:

$-m - 5m - 3m = -9m$

Combining the $n$ variable terms yields:

$-9n - 18n + 13n = -14n$

Combining the constant numerical terms yields:

$16 + 11 - 21 = 6$

Summing all combined components produces the final simplified polynomial:

$-9m - 14n + 6$

Square of a Binomial

Squaring a binomial expression follows set algebraic formulas based on whether the binomial is a sum or a difference.

The algebraic formula for the square of a sum binomial is:

$(a + b)^2 = a^2 + 2ab + b^2$

The algebraic formula for the square of a difference binomial is:

$(a - b)^2 = a^2 - 2ab + b^2$

Expanding any squared binomial requires a three-step procedure:

Step 1: Square the first term ($a^2$).

Step 2: Multiply the first term and the second term together, then multiply that product by $2$ ($2ab$).

Step 3: Square the last term ($b^2$).