Algebraic Expressions Cheatsheet

Cheatsheet Content

1. Basic Definitions Variable: A symbol (usually a letter) representing an unknown value (e.g., $x, y, a$). Constant: A fixed value that does not change (e.g., $5, -3, \pi$). Term: A single number, a single variable, or variables multiplied together (e.g., $4x, -7, 2xy^2$). Coefficient: The numerical part of a term that contains a variable (e.g., in $5x$, $5$ is the coefficient). Expression: A combination of terms joined by addition or subtraction (e.g., $3x + 2y - 7$). Equation: Two expressions set equal to each other (e.g., $3x + 2 = 11$). Inequality: Two expressions compared using $ , \le, \ge$ (e.g., $2x - 1 2. Types of Expressions Monomial: An expression with one term (e.g., $5x^2, -3xy, 7$). Binomial: An expression with two terms (e.g., $x + 5, 2y - 3z$). Trinomial: An expression with three terms (e.g., $x^2 + 2x - 1$). Polynomial: An expression with one or more terms, where variables have non-negative integer exponents (e.g., $4x^3 - 2x + 9$). Degree of a Term: The sum of the exponents of the variables in the term (e.g., degree of $3x^2y^3$ is $2+3=5$). Degree of a Polynomial: The highest degree of any term in the polynomial. 3. Operations with Expressions 3.1. Combining Like Terms Like terms have the same variables raised to the same powers. Only like terms can be added or subtracted. Example: $3x + 5x - 2y + y = (3+5)x + (-2+1)y = 8x - y$. 3.2. Distributive Property $a(b+c) = ab + ac$ Example: $2(x+3) = 2x + 6$. Example: $-3(y-4) = -3y + 12$. 3.3. Multiplying Polynomials Monomial by Polynomial: Distribute the monomial to each term. Example: $2x(x^2 - 3x + 1) = 2x^3 - 6x^2 + 2x$. Binomial by Binomial (FOIL): First, Outer, Inner, Last. $(a+b)(c+d) = ac + ad + bc + bd$. Example: $(x+2)(x-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$. Polynomial by Polynomial: Multiply each term of the first polynomial by each term of the second polynomial, then combine like terms. 4. Special Products Difference of Squares: $(a-b)(a+b) = a^2 - b^2$. Perfect Square Trinomials: $(a+b)^2 = a^2 + 2ab + b^2$. $(a-b)^2 = a^2 - 2ab + b^2$. Sum/Difference of Cubes: $(a+b)(a^2-ab+b^2) = a^3 + b^3$. $(a-b)(a^2+ab+b^2) = a^3 - b^3$. 5. Factoring Expressions Greatest Common Factor (GCF): Find the largest common factor of all terms. Example: $6x^2y - 9xy^2 = 3xy(2x - 3y)$. Factoring Trinomials ($ax^2 + bx + c$): If $a=1$: Find two numbers that multiply to $c$ and add to $b$. Example: $x^2 + 5x + 6 = (x+2)(x+3)$. If $a \ne 1$: Use grouping or the "AC method". Example: $2x^2 + 7x + 3 = 2x^2 + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3)$. Difference of Squares: $a^2 - b^2 = (a-b)(a+b)$. Example: $4x^2 - 9 = (2x - 3)(2x + 3)$. Perfect Square Trinomials: $a^2 + 2ab + b^2 = (a+b)^2$. $a^2 - 2ab + b^2 = (a-b)^2$. Factoring by Grouping: For 4 or more terms. Example: $x^3 + 2x^2 + 3x + 6 = x^2(x+2) + 3(x+2) = (x^2+3)(x+2)$. 6. Exponent Rules Product Rule: $x^m \cdot x^n = x^{m+n}$. Quotient Rule: $\frac{x^m}{x^n} = x^{m-n}$ (for $x \ne 0$). Power Rule: $(x^m)^n = x^{mn}$. Power of a Product: $(xy)^n = x^n y^n$. Power of a Quotient: $(\frac{x}{y})^n = \frac{x^n}{y^n}$ (for $y \ne 0$). Zero Exponent: $x^0 = 1$ (for $x \ne 0$). Negative Exponent: $x^{-n} = \frac{1}{x^n}$ (for $x \ne 0$). Fractional Exponent: $x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$. 7. Radical Expressions Definition: $\sqrt[n]{a}$ is the $n$-th root of $a$. Product Rule: $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. Quotient Rule: $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ (for $b \ne 0$). Simplifying Radicals: Factor out perfect squares/cubes etc. Example: $\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$. Adding/Subtracting: Only like radicals (same index and radicand) can be combined. Example: $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$. Rationalizing the Denominator: Eliminate radicals from the denominator. If $\sqrt{a}$: Multiply by $\frac{\sqrt{a}}{\sqrt{a}}$. If $a+\sqrt{b}$: Multiply by the conjugate $\frac{a-\sqrt{b}}{a-\sqrt{b}}$. 8. Solving Equations Linear Equations: Isolate the variable. Example: $2x + 5 = 11 \implies 2x = 6 \implies x = 3$. Quadratic Equations ($ax^2+bx+c=0$): Factoring: Set each factor to zero. Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Completing the Square: Convert to $(x+k)^2 = d$. Equations with Radicals: Isolate the radical, then raise both sides to the power of the index. Check for extraneous solutions. Equations with Fractions: Multiply by the Least Common Denominator (LCD) to clear denominators. Check for values that make denominators zero.