Mathematics Algebra Syllabus

Algebra syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Algebra in Mathematics, not a summary of it.

1. 1

   Linear Equations

   overview

   Examined as a single unit within Algebra — no further topic split in the official outline.

2. 2

   Quadratic Equations

   overview

   Examined as a single unit within Algebra — no further topic split in the official outline.

3. 3

   Polynomials

   overview

   Examined as a single unit within Algebra — no further topic split in the official outline.

4. 4

   Factoring

   overview

   Examined as a single unit within Algebra — no further topic split in the official outline.

5. 5

   Inequalities

   overview

   Examined as a single unit within Algebra — no further topic split in the official outline.

6. 6

   Functions

   overview

   Examined as a single unit within Algebra — no further topic split in the official outline.

7. 7

   Rational Expressions

   overview

   Examined as a single unit within Algebra — no further topic split in the official outline.

8. 8

   Exponents and Radicals

   overview

   Examined as a single unit within Algebra — no further topic split in the official outline.

9. 9

   Logarithms

   overview

   Examined as a single unit within Algebra — no further topic split in the official outline.

Algebra flashcards for Mathematics

19 of 50 cards from the Algebra deck — real questions with worked answers.

1. What is a monomial, and how does it differ from a polynomial?

   A monomial is a single term consisting of a product of constants and variables with non-negative integer exponents (e.g., $5x^{2}y$). A polynomial is a sum of one or more monomials (terms), e.g., $3x^{2} + 2x - 7$.

2. State the general form of a quadratic equation and the quadratic formula for its roots.

   General form: $ax^{2} + bx + c = 0$ with $a \neq 0$. Roots: $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$

3. What is the discriminant of a quadratic, and what does each case tell you about the roots?

   The discriminant is $\Delta = b^{2} - 4ac$. If $\Delta > 0$: two distinct real roots; if $\Delta = 0$: one repeated real root; if $\Delta < 0$: two complex conjugate roots.

4. State Vieta's formulas relating the roots $r_1, r_2$ of $ax^{2}+bx+c=0$ to its coefficients.

   Sum of roots: $r_1 + r_2 = -\dfrac{b}{a}$. Product of roots: $r_1 r_2 = \dfrac{c}{a}$.

5. Expand $(a+b)^{2}$, $(a-b)^{2}$, and $(a+b)(a-b)$.

   $(a+b)^{2} = a^{2} + 2ab + b^{2}$; $(a-b)^{2} = a^{2} - 2ab + b^{2}$; $(a+b)(a-b) = a^{2} - b^{2}$.

6. Give the formulas for the sum and difference of two cubes.

   $a^{3} + b^{3} = (a+b)(a^{2} - ab + b^{2})$ and $a^{3} - b^{3} = (a-b)(a^{2} + ab + b^{2})$.

7. State the product, quotient, and power rules for exponents.

   Product: $a^{m} \cdot a^{n} = a^{m+n}$. Quotient: $\dfrac{a^{m}}{a^{n}} = a^{m-n}$. Power: $(a^{m})^{n} = a^{mn}$.

8. What do a zero exponent and a negative exponent equal?

   $a^{0} = 1$ (for $a \neq 0$) and $a^{-n} = \dfrac{1}{a^{n}}$ (for $a \neq 0$).

9. How is a fractional (rational) exponent expressed using radicals?

   $a^{m/n} = \sqrt[n]{a^{m}} = \left(\sqrt[n]{a}\right)^{m}$.

10. State the three laws of logarithms (product, quotient, power).

    $\log_b(xy) = \log_b x + \log_b y$; $\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y$; $\log_b(x^{p}) = p\,\log_b x$.

11. State the change-of-base formula for logarithms.

    $$\log_b a = \frac{\log_c a}{\log_c b}$$ for any valid base $c$.

12. What logarithmic statement is equivalent to $b^{y} = x$?

    $\log_b x = y$, where $b > 0$, $b \neq 1$, and $x > 0$.

13. What is the slope-intercept form of a line, and what do its parameters represent?

    $y = mx + c$, where $m$ is the slope (rate of change) and $c$ is the $y$-intercept (value of $y$ when $x = 0$).

14. Give the formula for the slope of a line through points $(x_1,y_1)$ and $(x_2,y_2)$.

    $$m = \frac{y_2 - y_1}{x_2 - x_1}, \quad x_1 \neq x_2$$

15. What is the point-slope form of a linear equation?

    $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

16. How do the slopes of parallel and perpendicular lines compare?

    Parallel lines have equal slopes: $m_1 = m_2$. Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$.

17. What is a system of linear equations, and what are its three possible solution types?

    A set of two or more linear equations in the same variables. It can have exactly one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines).

18. Describe the substitution method for solving a system of equations.

    Solve one equation for one variable, substitute that expression into the other equation to get a single-variable equation, solve it, then back-substitute to find the remaining variable.

19. Describe the elimination (addition) method for solving a linear system.

    Multiply equations as needed so one variable has opposite coefficients, add the equations to eliminate that variable, solve for the remaining one, then back-substitute.

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