ALGEBRA HELP SHEET - Video Review Flashcards

Cards cover real numbers, polynomial operations, factoring, properties, solving equations, absolute value inequalities, lines, logarithms, and basic analytic geometry.

What are Natural Numbers?

1, 2, 3, 4, 5, 6, …

What are Whole Numbers?

0, 1, 2, 3, 4, 5, 6, …

What are Integers?

-3, -2, -1, 0, 1, 2, 3, …

What are Rational Numbers?

Numbers that can be written as a/b where a and b are integers and b ≠ 0.

What are Irrational Numbers?

Decimals that do not terminate or repeat (e.g., √3, π).

What is a Real Number?

The union of rational and irrational numbers.

What is an nth root of x?

√[n]{x}, the real nth root of x for a positive integer n > 1.

Expand (a+b)^2.

a^2 + 2ab + b^2.

Expand (a-b)^2.

a^2 - 2ab + b^2.

Expand (a+b)^3.

a^3 + 3a^2b + 3ab^2 + b^3.

Expand (a-b)^3.

a^3 - 3a^2b + 3ab^2 - b^3.

What is the Distributive Property?

a(b+c) = ab + ac (and (a+b)c = ac + bc).

What is the Difference of Squares?

a^2 - b^2 = (a+b)(a-b).

What is the Reciprocal (multiplicative inverse) of a number a (a ≠ 0)?

1/a, since a·(1/a) = 1.

What is the Additive Inverse of a number a?

-a, since a + (-a) = 0.

What is the Additive Identity?

0.

What is the Multiplicative Identity?

1.

What is the Double-Negative Rule?

-(-b) = b.

What does the Zero Factor Theorem state?

If ab = 0, then a = 0 or b = 0.

What is the Absolute Value inequality |x| < k (k > 0) equivalent to?

-k < x < k.

What is the Absolute Value inequality |x| > k (k > 0) equivalent to?

x < -k or x > k.

What is the Linear Inequality rule for adding the same number?

If c is real and c > 0, then a+c < b+c whenever a < b; if c < 0, the inequality reverses.

What is the Slope of a Line?

m = (y2 - y1) / (x2 - x1).

What is a horizontal line’s slope?

0.

What is a vertical line’s slope?

Undefined (slope is not a real number).

What is the Logarithm Base Definition?

log_b(x) is the exponent to which base b must be raised to get x (x > 0, b > 0, b ≠ 1).

What are key values of logarithms?

logb(b) = 1 and logb(1) = 0.

What is the Product Rule for logarithms?

logb(MN) = logb(M) + log_b(N).

What is the Quotient Rule for logarithms?

logb(M/N) = logb(M) - log_b(N).

What is the Power Rule for logarithms?

logb(M^k) = k · logb(M).

What is the Change of Base Formula?

logb(x) = logk(x) / log_k(b) for any base k > 0, k ≠ 1.

What is the inverse relationship between logarithms and exponentials?

y = log_b(x) is the inverse of y = b^x.

What is the Slope-Intercept Form of a Line?

y = mx + b, where m is the slope and b is the y-intercept.

What is the Point-Slope Form of a Line?

y - y1 = m(x - x1), where (x1, y1) is a point on the line.

What is the General Form of a Line?

Ax + By = C.

What is the equation of a horizontal line?

y = constant.

What is the equation of a vertical line?

x = constant.

What is the Distance Formula between P(x1,y1) and Q(x2,y2)?

d = √[(x2 - x1)^2 + (y2 - y1)^2].

What is the Pythagorean Theorem?

For a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse.

What is the Quadratic Formula?

For ax^2 + bx + c = 0, x = [-b ± √(b^2 - 4ac)] / (2a) (a ≠ 0).

What is the Discriminant and its cases?

Δ = b^2 - 4ac; Δ > 0: two real roots; Δ = 0: one real root; Δ < 0: nonreal roots.

What is Completing the Square?

Add (b/2a)^2 to x^2 + bx to form (x + b/(2a))^2.

What is the Distance formula derived from the Pythagorean Theorem?

d = √[(x2 - x1)^2 + (y2 - y1)^2].

What is the equation of a line through two points (x1,y1) and (x2,y2) in slope form?

m = (y2 - y1)/(x2 - x1); then use y - y1 = m(x - x1).