Mathematics Algebra Syllabus

Give the formula for the sum of the first $n$ terms of an arithmetic series.

$$S_n = \frac{n}{2}\big(a_1 + a_n\big) = \frac{n}{2}\big(2a_1 + (n-1)d\big)$$

What is the formula for the $n$-th term of a geometric sequence?

$a_n = a_1 \cdot r^{\,n-1}$, where $a_1$ is the first term and $r$ is the common ratio.

Give the formula for the sum of the first $n$ terms of a geometric series.

$$S_n = a_1 \cdot \frac{1 - r^{n}}{1 - r}, \quad r \neq 1$$

What is the sum of an infinite geometric series, and when does it converge?

$$S_\infty = \frac{a_1}{1 - r}, \quad \text{valid for } |r| < 1.$$ If $|r| \geq 1$ the series diverges.

State the binomial theorem for $(a+b)^{n}$.

$$(a+b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}, \quad \binom{n}{k} = \frac{n!}{k!\,(n-k)!}$$

How do you complete the square for $x^{2} + bx$?

Add and subtract $\left(\dfrac{b}{2}\right)^{2}$: $x^{2} + bx = \left(x + \dfrac{b}{2}\right)^{2} - \left(\dfrac{b}{2}\right)^{2}$.

What is the domain of a function, and what is its range?

The domain is the set of all permissible input values ($x$); the range is the set of all resulting output values ($y$ or $f(x)$).

What defines a function, and what test checks whether a graph is a function?

A function assigns exactly one output to each input. The vertical line test: if any vertical line crosses the graph more than once, it is not a function.

What is the absolute value $|x|$, and how do you solve $|x| = a$ for $a > 0$?

$|x|$ is the distance of $x$ from $0$: $|x| = x$ if $x \geq 0$, and $|x| = -x$ if $x < 0$. Solving $|x| = a$ gives $x = a$ or $x = -a$.

When you multiply or divide both sides of an inequality by a negative number, what must you do?

Reverse the direction of the inequality sign. E.g., from $-2x < 6$ you get $x > -3$.

What are the vertex form of a parabola and the coordinates of its vertex?

Vertex form: $y = a(x-h)^{2} + k$, with vertex at $(h, k)$. It opens upward if $a > 0$ and downward if $a < 0$.

Give the formula for the $x$-coordinate of the vertex of $y = ax^{2}+bx+c$.

$x = -\dfrac{b}{2a}$; this is also the axis of symmetry. The $y$-coordinate is found by substituting this value back into the equation.

State the Factor Theorem.

For a polynomial $P(x)$, $(x - a)$ is a factor of $P(x)$ if and only if $P(a) = 0$ (i.e., $a$ is a root).

State the Remainder Theorem.

When a polynomial $P(x)$ is divided by $(x - a)$, the remainder equals $P(a)$.

What is a rational expression, and what value(s) must be excluded?

A rational expression is a ratio of two polynomials $\dfrac{P(x)}{Q(x)}$. Any value making the denominator zero ($Q(x) = 0$) must be excluded from the domain.

How do you add two fractions $\dfrac{a}{b} + \dfrac{c}{d}$?

Use a common denominator: $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

What is a direct proportion versus an inverse proportion?

Direct: $y = kx$, so $y$ increases as $x$ increases ($k$ constant). Inverse: $y = \dfrac{k}{x}$, so $y$ decreases as $x$ increases.

State the formula for the determinant of a $2\times 2$ matrix.

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det(A) = ad - bc$.

What is the inverse of a $2\times 2$ matrix, and when does it exist?

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $$A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix},$$ which exists only when $\det(A) = ad - bc \neq 0$.

State the commutative, associative, and distributive properties of real numbers.

Commutative: $a+b = b+a$, $ab = ba$. Associative: $(a+b)+c = a+(b+c)$, $(ab)c = a(bc)$. Distributive: $a(b+c) = ab + ac$.

What are the additive identity, multiplicative identity, and their inverses?

Additive identity: $0$ (since $a+0=a$); its inverse is $-a$. Multiplicative identity: $1$ (since $a\cdot 1=a$); its inverse is $\dfrac{1}{a}$ for $a\neq 0$.

How do you simplify a square root such as $\sqrt{50}$ (extracting factors)?

Factor out perfect squares: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$. Use $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$.

How do you rationalize the denominator of $\dfrac{1}{\sqrt{a}}$?

Multiply numerator and denominator by $\sqrt{a}$: $\dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}$.

What is function composition $(f \circ g)(x)$, and is it commutative?

$(f \circ g)(x) = f(g(x))$: apply $g$ first, then $f$. It is generally not commutative, so $f(g(x)) \neq g(f(x))$ in general.

How do you find the inverse function $f^{-1}(x)$ of a one-to-one function?

Replace $f(x)$ with $y$, swap $x$ and $y$, then solve for $y$. The result is $f^{-1}(x)$, satisfying $f(f^{-1}(x)) = x$.

What is the standard form of the equation of a circle with center $(h,k)$ and radius $r$?

$(x - h)^{2} + (y - k)^{2} = r^{2}$.

Give the distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$.

$$d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$$

Give the midpoint formula for the segment joining $(x_1,y_1)$ and $(x_2,y_2)$.

$$M = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right)$$

Classify polynomials by degree: name degrees 0 through 3.

Degree 0: constant; degree 1: linear; degree 2: quadratic; degree 3: cubic. (Degree 4 is quartic, degree 5 quintic.)

What does the Fundamental Theorem of Algebra state?

Every non-constant polynomial of degree $n$ with complex coefficients has exactly $n$ roots in the complex numbers, counting multiplicity.