Title: Algebra Fundamentals — LearnCoachAssist

Q: What is a variable in algebra?
A: A variable is a symbol, usually a letter like x or y, that represents an unknown or changeable quantity in a mathematical expression or equation.

Q: What is an algebraic expression?
A: An algebraic expression is a combination of variables, constants, and operations (such as addition, subtraction, multiplication, and division) without an equals sign. For example, 3x + 5 is an algebraic expression.

Q: What is a coefficient?
A: A coefficient is the numerical factor multiplied by a variable in a term. In the term 7x, the coefficient is 7.

Q: What is a constant in an algebraic expression?
A: A constant is a fixed numerical value that does not change. In the expression 2x + 9, the constant is 9.

Q: What does it mean to evaluate an expression?
A: To evaluate an expression means to substitute specific values for the variables and then simplify using the order of operations to obtain a numerical result.

Q: What are like terms?
A: Like terms are terms that have the same variable raised to the same power. For example, 3x² and −5x² are like terms because they both contain x². Like terms can be combined by adding or subtracting their coefficients.

Q: What is the distributive property?
A: The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by each term inside a set of parentheses.

Q: Simplify the expression: 4(2x − 3) + 5x
A: Distribute first: 8x − 12 + 5x. Then combine like terms: 13x − 12.

Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable is 1. Its graph is a straight line. The general form is ax + b = 0.

Q: Solve for x: 3x + 7 = 22
A: Subtract 7 from both sides: 3x = 15. Divide both sides by 3: x = 5.

Q: Solve for x: 5(x − 2) = 3x + 4
A: Distribute: 5x − 10 = 3x + 4. Subtract 3x: 2x − 10 = 4. Add 10: 2x = 14. Divide by 2: x = 7.

Q: What does it mean for an equation to have no solution?
A: An equation has no solution when simplifying leads to a contradiction such as 0 = 5. This means no value of the variable can make the equation true.

Q: What does it mean for an equation to have infinitely many solutions?
A: An equation has infinitely many solutions when simplifying leads to a tautology such as 0 = 0. Every real number satisfies the equation.

Q: Solve for x: 2(x + 3) = 2x + 6
A: Distribute: 2x + 6 = 2x + 6. This simplifies to 0 = 0, which is always true. The equation has infinitely many solutions.

Q: How do you solve a linear equation with fractions?
A: Multiply every term by the least common denominator (LCD) to clear all fractions, then solve the resulting equation using standard methods.

Q: Solve for x: (x/3) + 2 = 5
A: Subtract 2 from both sides: x/3 = 3. Multiply both sides by 3: x = 9.

Q: What is a quadratic equation?
A: A quadratic equation is a polynomial equation of degree 2, written in standard form as ax² + bx + c = 0 where a ≠ 0.

Q: What is the quadratic formula?
A: The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). It gives the solutions to any quadratic equation in the form ax² + bx + c = 0.

Q: What is the discriminant of a quadratic equation?
A: The discriminant is b² − 4ac. If it is positive, there are two distinct real solutions. If zero, there is exactly one real solution. If negative, there are two complex (non-real) solutions.

Q: Solve by factoring: x² − 5x + 6 = 0
A: Factor: (x − 2)(x − 3) = 0. Set each factor to zero: x = 2 or x = 3.

Q: Solve using the quadratic formula: 2x² + 3x − 2 = 0
A: Here a=2, b=3, c=−2. Discriminant = 9 + 16 = 25. So x = (−3 ± 5)/4, giving x = 1/2 or x = −2.

Q: What is completing the square?
A: Completing the square is a technique for rewriting a quadratic expression ax² + bx + c in the form a(x − h)² + k. You take half of the coefficient of x, square it, and add and subtract that value.

Q: Solve by completing the square: x² + 6x + 5 = 0
A: Rewrite: x² + 6x = −5. Add (6/2)² = 9 to both sides: (x + 3)² = 4. Take square root: x + 3 = ±2, so x = −1 or x = −5.

Q: What is the vertex form of a quadratic equation?
A: The vertex form is y = a(x − h)² + k, where (h, k) is the vertex of the parabola. If a > 0 the parabola opens upward; if a < 0 it opens downward.

Q: How many solutions can a quadratic equation have?
A: A quadratic equation can have two distinct real solutions, one repeated real solution, or two complex conjugate solutions, depending on the value of the discriminant.

Q: What is the sum and product of the roots of ax² + bx + c = 0?
A: By Vieta's formulas, the sum of the roots is −b/a and the product of the roots is c/a.

Q: What is a system of equations?
A: A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously.

Q: What are the three methods for solving a system of linear equations?
A: The three main methods are graphing (finding intersection points), substitution (solving one equation for a variable and substituting), and elimination (adding or subtracting equations to cancel a variable).

Q: Solve by substitution: y = 2x + 1 and 3x + y = 11
A: Substitute y: 3x + (2x + 1) = 11. Simplify: 5x + 1 = 11, so x = 2. Then y = 2(2) + 1 = 5. Solution: (2, 5).

Q: Solve by elimination: 2x + 3y = 12 and 4x − 3y = 6
A: Add the equations: 6x = 18, so x = 3. Substitute back: 2(3) + 3y = 12, so 3y = 6 and y = 2. Solution: (3, 2).

Q: What does it mean when a system of equations is inconsistent?
A: An inconsistent system has no solution. Graphically, this means the lines are parallel and never intersect. The equations represent contradictory conditions.

Q: What does it mean when a system of equations is dependent?
A: A dependent system has infinitely many solutions. The equations represent the same line, so every point on that line is a solution.

Q: How can you determine the number of solutions of a 2×2 linear system by comparing slopes?
A: If the slopes are different, there is exactly one solution. If the slopes are equal and y-intercepts are different, there is no solution. If slopes and y-intercepts are both equal, there are infinitely many solutions.

Q: What is Cramer's Rule for a 2×2 system?
A: For the system ax + by = e and cx + dy = f, Cramer's Rule gives x = (ed − bf)/(ad − bc) and y = (af − ce)/(ad − bc), provided the determinant ad − bc ≠ 0.

Q: What is the difference between a strict and non-strict inequality?
A: A strict inequality uses < or > (the boundary is not included), while a non-strict inequality uses ≤ or ≥ (the boundary is included).

Q: How do you graph a linear inequality in two variables?
A: Graph the boundary line (dashed for strict, solid for non-strict). Then test a point not on the line to determine which half-plane to shade.

Q: What happens when you multiply or divide both sides of an inequality by a negative number?
A: You must reverse the direction of the inequality sign. For example, if −2x > 6, dividing by −2 gives x < −3.

Q: Solve: 3x − 4 ≤ 11
A: Add 4: 3x ≤ 15. Divide by 3: x ≤ 5. In interval notation, the solution is (−∞, 5].

Q: Solve the compound inequality: −2 < 3x + 1 ≤ 10
A: Subtract 1 from all parts: −3 < 3x ≤ 9. Divide by 3: −1 < x ≤ 3. In interval notation: (−1, 3].

Q: Solve: 2x + 5 > 3x − 1
A: Subtract 2x: 5 > x − 1. Add 1: 6 > x, or equivalently x < 6. In interval notation: (−∞, 6).

Q: What is interval notation?
A: Interval notation is a way to describe a set of numbers using parentheses and brackets. Parentheses ( indicate the endpoint is excluded, and brackets [ indicate the endpoint is included. For example, [2, 5) means all numbers from 2 (included) to 5 (excluded).

Q: What is a polynomial?
A: A polynomial is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where all exponents are non-negative integers. For example, 4x³ − 2x + 7.

Q: What is the degree of a polynomial?
A: The degree is the highest power of the variable in the polynomial. For example, 5x⁴ + 3x² − 1 has degree 4.

Q: How do you add or subtract polynomials?
A: Combine like terms — terms with the same variable raised to the same power. Add or subtract their coefficients accordingly.

Q: How do you multiply two polynomials?
A: Multiply each term of the first polynomial by every term of the second polynomial (using the distributive property), then combine like terms. For binomials, you can use the FOIL method.

Q: What is the FOIL method?
A: FOIL stands for First, Outer, Inner, Last. It is a shortcut for multiplying two binomials: (a + b)(c + d) = ac + ad + bc + bd.

Q: Expand: (2x + 3)(x − 4)
A: Using FOIL: First: 2x², Outer: −8x, Inner: 3x, Last: −12. Combine: 2x² − 5x − 12.

Q: What is the Remainder Theorem?
A: The Remainder Theorem states that when a polynomial f(x) is divided by (x − c), the remainder equals f(c). If f(c) = 0, then (x − c) is a factor.

Q: What is the greatest common factor (GCF) in factoring?
A: The GCF is the largest expression that divides evenly into all terms of a polynomial. Always look for and factor out the GCF as the first step in factoring.

Q: Factor: 6x³ + 9x²
A: The GCF is 3x². Factor it out: 3x²(2x + 3).

Q: What is the difference of squares pattern?
A: The difference of squares states a² − b² = (a + b)(a − b). It applies when you have two perfect squares separated by subtraction.

Q: Factor: x² − 16
A: This is a difference of squares: x² − 4² = (x + 4)(x − 4).

Q: What is a perfect square trinomial?
A: A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². The middle term is always twice the product of the square roots of the first and last terms.

Q: Factor: x² + 10x + 25
A: This is a perfect square trinomial: (x + 5)², since 5² = 25 and 2 · x · 5 = 10x.

Q: How do you factor a trinomial of the form x² + bx + c?
A: Find two numbers that multiply to c and add to b. If those numbers are p and q, then x² + bx + c = (x + p)(x + q).

Q: What is the sum or difference of cubes formula?
A: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). Remember the mnemonic: SOAP (Same, Opposite, Always Positive) for the signs.

Q: What is the product rule for exponents?
A: When multiplying powers with the same base, add the exponents: aᵐ · aⁿ = aᵐ⁺ⁿ.

Q: What is the power rule for exponents?
A: When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ.

Q: What is the quotient rule for exponents?
A: When dividing powers with the same base, subtract the exponents: aᵐ / aⁿ = aᵐ⁻ⁿ, provided a ≠ 0.

Q: What does a zero exponent mean?
A: Any nonzero number raised to the zero power equals 1: a⁰ = 1 (where a ≠ 0). This follows from the quotient rule: aⁿ / aⁿ = aⁿ⁻ⁿ = a⁰ = 1.

Q: What does a negative exponent mean?
A: A negative exponent indicates a reciprocal: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8.

Q: How do you convert between radical and exponential notation?
A: The nth root of a can be written as a fractional exponent: ⁿ√a = a^(1/n). More generally, ⁿ√(aᵐ) = a^(m/n).

Q: Simplify: √50
A: Factor 50 as 25 × 2. Then √50 = √25 · √2 = 5√2.

Q: How do you rationalize a denominator containing a single radical?
A: Multiply both the numerator and denominator by the radical in the denominator. For example, 1/√3 = (1 · √3)/(√3 · √3) = √3/3.

Q: What is a function?
A: A function is a relation that assigns exactly one output value for every input value. If f(x) = x², each value of x produces exactly one value of f(x).

Q: What is the domain of a function?
A: The domain is the set of all permissible input values (x-values) for which the function is defined. Common restrictions include division by zero and even roots of negative numbers.

Q: What is the range of a function?
A: The range is the set of all possible output values (y-values) that the function can produce.

Q: How do you find the domain of f(x) = 1/(x − 3)?
A: Set the denominator not equal to zero: x − 3 ≠ 0, so x ≠ 3. The domain is all real numbers except 3, written as (−∞, 3) ∪ (3, ∞).

Q: How do you find the domain of f(x) = √(x − 2)?
A: The expression under the square root must be non-negative: x − 2 ≥ 0, so x ≥ 2. The domain is [2, ∞).

Q: What is function composition?
A: Function composition combines two functions by using the output of one as the input of the other. It is written as (f ∘ g)(x) = f(g(x)). You first evaluate g(x), then apply f to that result.

Q: If f(x) = 2x + 1 and g(x) = x², what is (f ∘ g)(3)?
A: First find g(3) = 9. Then f(9) = 2(9) + 1 = 19. So (f ∘ g)(3) = 19.

Q: What is an inverse function?
A: An inverse function f⁻¹ reverses the action of f. If f(a) = b, then f⁻¹(b) = a. Graphically, the inverse is a reflection of f over the line y = x.

Q: How do you find the inverse of a function algebraically?
A: Replace f(x) with y, swap x and y, then solve for y. The resulting expression is f⁻¹(x). Always verify by checking that f(f⁻¹(x)) = x.

Q: What is the vertical line test?
A: The vertical line test states that a graph represents a function if and only if every vertical line intersects the graph at most once. If any vertical line crosses the graph more than once, the relation is not a function.

Q: What is a piecewise function?
A: A piecewise function is defined by different expressions for different intervals of the domain. For example, f(x) = x² if x ≥ 0 and f(x) = −x if x < 0.

Q: What is a logarithm?
A: A logarithm is the inverse of exponentiation. log_b(x) = y means bʸ = x. It answers the question: "To what power must the base b be raised to produce x?"

Q: What is the common logarithm?
A: The common logarithm is the logarithm with base 10, written as log(x) or log₁₀(x). For example, log(1000) = 3 because 10³ = 1000.

Q: What is the natural logarithm?
A: The natural logarithm has base e (approximately 2.718) and is written as ln(x). It is widely used in calculus and exponential growth/decay models.

Q: What is the product rule for logarithms?
A: log_b(MN) = log_b(M) + log_b(N). The log of a product equals the sum of the logs of the factors.

Q: What is the quotient rule for logarithms?
A: log_b(M/N) = log_b(M) − log_b(N). The log of a quotient equals the difference of the logs.

Q: What is the power rule for logarithms?
A: log_b(Mⁿ) = n · log_b(M). The log of a power equals the exponent times the log of the base expression.

Q: What is the change of base formula?
A: log_b(x) = log_a(x) / log_a(b). This allows you to compute logarithms in any base using a calculator that only has common or natural log keys: log_b(x) = ln(x) / ln(b).

Q: Solve for x: log₂(x) = 5
A: Convert to exponential form: x = 2⁵ = 32.

Q: What is an arithmetic sequence?
A: An arithmetic sequence is a sequence where each term differs from the previous one by a constant called the common difference (d). The nth term is aₙ = a₁ + (n − 1)d.

Q: What is the formula for the sum of the first n terms of an arithmetic series?
A: Sₙ = n(a₁ + aₙ)/2, or equivalently Sₙ = n/2 · [2a₁ + (n − 1)d]. It is the number of terms times the average of the first and last terms.

Q: What is a geometric sequence?
A: A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant called the common ratio (r). The nth term is aₙ = a₁ · rⁿ⁻¹.

Q: What is the formula for the sum of the first n terms of a geometric series?
A: Sₙ = a₁(1 − rⁿ) / (1 − r), provided r ≠ 1.

Q: When does an infinite geometric series converge, and what is its sum?
A: An infinite geometric series converges when |r| < 1. Its sum is S = a₁ / (1 − r).

Q: Find the 10th term of the arithmetic sequence: 3, 7, 11, 15, …
A: The common difference is d = 4 and a₁ = 3. Using aₙ = a₁ + (n−1)d: a₁₀ = 3 + 9(4) = 39.

Q: Find the sum of the infinite geometric series: 8 + 4 + 2 + 1 + …
A: Here a₁ = 8 and r = 1/2. Since |r| < 1, the sum is S = 8 / (1 − 1/2) = 8 / (1/2) = 16.

Q: What does the absolute value of a number represent?
A: The absolute value |a| represents the distance of a from zero on the number line. It is always non-negative: |a| ≥ 0 for all real numbers a.

Q: How do you solve an equation of the form |x| = a where a ≥ 0?
A: Split into two cases: x = a or x = −a. If a < 0, the equation has no solution because absolute value cannot be negative.

Q: Solve: |2x − 3| = 7
A: Case 1: 2x − 3 = 7 → x = 5. Case 2: 2x − 3 = −7 → x = −2. The solutions are x = 5 and x = −2.

Q: How do you solve |x| < a (where a > 0)?
A: Rewrite as a compound inequality: −a < x < a. The solution is the open interval (−a, a).

Q: How do you solve |x| > a (where a > 0)?
A: Split into two separate inequalities: x > a or x < −a. The solution in interval notation is (−∞, −a) ∪ (a, ∞).

Q: What is a rational expression?
A: A rational expression is a fraction in which the numerator and denominator are both polynomials, such as (x² + 1) / (x − 3). It is undefined wherever the denominator equals zero.

Q: How do you simplify a rational expression?
A: Factor both the numerator and denominator completely, then cancel any common factors. Always state the restrictions on the variable (values that make the original denominator zero).

Q: How do you multiply two rational expressions?
A: Multiply the numerators together and the denominators together: (A/B) · (C/D) = AC / BD. Factor and cancel common factors before multiplying to simplify.

Q: How do you divide two rational expressions?
A: Multiply by the reciprocal of the divisor: (A/B) ÷ (C/D) = (A/B) · (D/C) = AD / BC. Then simplify by canceling common factors.

Q: How do you add or subtract rational expressions with different denominators?
A: Find the least common denominator (LCD), rewrite each fraction with the LCD, then add or subtract the numerators. Finally, simplify the resulting expression.