Mathematics Admissions Test (MAT) Algebra and Equations Syllabus

Algebra and Equations syllabus — full chapter and topic list

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

1. 1

   Manipulation and Surds

   3 topics

   • Algebraic Manipulation
     • Expanding brackets and the binomial expansion for positive integer powers
     • Factorising quadratics and cubics by inspection and grouping
     • Algebraic fractions: addition, simplification and partial-fraction intuition
     • Rearranging formulae to change the subject
   • Indices and Surds
     • Laws of indices including fractional and negative exponents
     • Simplifying surds and rationalising denominators
     • Evaluating expressions without a calculator
   • Completing the Square
     • Vertex form and identifying maxima and minima
     • Using completed-square form to prove non-negativity
2. 2

   Quadratic and Polynomial Equations

   3 topics

   • Solving Quadratics
     • Factorisation, formula and completing the square
     • The discriminant and the nature of roots
     • Sum and product of roots (Vieta's relations)
   • Disguised Quadratics
     • Quadratics in a substituted variable (e.g. in x squared or in 2 to the x)
     • Equations reducible to quadratics by substitution
   • Conditions on Coefficients
     • Finding ranges of a parameter for real, equal or no roots
     • Ensuring positivity of a quadratic for all real x
3. 3

   Simultaneous Equations and Inequalities

   3 topics

   • Simultaneous Equations
     • Linear systems by elimination and substitution
     • One linear, one quadratic system and geometric interpretation
   • Linear and Quadratic Inequalities
     • Solving and representing solution sets on a number line
     • Sign analysis of factorised expressions
   • Inequalities with Algebraic Reasoning
     • Inequalities involving fractions and the need to consider sign of denominator
     • Establishing bounds and using the discriminant for inequality proofs

Algebra and Equations flashcards for Mathematics Admissions Test (MAT)

21 of 49 cards from the Algebra and Equations deck — real questions with worked answers.

1. What are the three laws of indices for multiplication, division, and powers of powers?

   $a^{m} \cdot a^{n} = a^{m+n}$, $\dfrac{a^{m}}{a^{n}} = a^{m-n}$, and $\left(a^{m}\right)^{n} = a^{mn}$.

2. What is the value of $a^{0}$ for any $a \neq 0$, and why?

   $a^{0} = 1$. It follows from $\dfrac{a^{n}}{a^{n}} = a^{n-n} = a^{0} = 1$.

3. How do you rewrite a negative index $a^{-n}$ as a positive index?

   $a^{-n} = \dfrac{1}{a^{n}}$ (the reciprocal of $a^{n}$).

4. What does a fractional index $a^{\frac{m}{n}}$ mean in terms of roots and powers?

   $a^{\frac{m}{n}} = \sqrt[n]{a^{m}} = \left(\sqrt[n]{a}\right)^{m}$. The denominator is the root and the numerator is the power.

5. What are the two key surd rules for multiplying and dividing roots?

   $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$ and $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$ (for $a, b \geq 0$, $b \neq 0$).

6. How do you rationalise the denominator of $\dfrac{1}{\sqrt{a}}$?

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

7. How do you rationalise a denominator of the form $\dfrac{1}{a + \sqrt{b}}$?

   Multiply top and bottom by the conjugate $a - \sqrt{b}$: $\dfrac{1}{a+\sqrt{b}} = \dfrac{a-\sqrt{b}}{a^{2}-b}$.

8. What is a conjugate surd, and what is special about the product of conjugates?

   The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$. Their product $\left(a+\sqrt{b}\right)\left(a-\sqrt{b}\right) = a^{2} - b$ is rational (no surds).

9. What is the difference of two squares factorisation?

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

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

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

11. What are the sum and difference of two cubes factorisations?

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

12. What is the general form of completing the square for $x^{2} + bx + c$?

    $x^{2} + bx + c = \left(x + \dfrac{b}{2}\right)^{2} + c - \dfrac{b^{2}}{4}$.

13. How do you complete the square when the leading coefficient $a \neq 1$, i.e. for $ax^{2}+bx+c$?

    Factor $a$ out of the first two terms first: $a\left(x^{2} + \dfrac{b}{a}x\right) + c = a\left(x + \dfrac{b}{2a}\right)^{2} + c - \dfrac{b^{2}}{4a}$.

14. In completed-square form $a(x-p)^{2}+q$, what do $p$ and $q$ tell you about the parabola?

    The vertex (turning point) is at $(p, q)$. The axis of symmetry is $x = p$, and $q$ is the minimum value (if $a>0$) or maximum (if $a<0$).

15. State the quadratic formula for the roots of $ax^{2} + bx + c = 0$.

    $x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$.

16. What is the discriminant of a quadratic, and what symbol is used?

    The discriminant is $\Delta = b^{2} - 4ac$, the expression under the square root in the quadratic formula.

17. How does the sign of the discriminant $\Delta = b^{2}-4ac$ classify the roots of a real quadratic?

    If $\Delta > 0$: two distinct real roots; if $\Delta = 0$: one repeated (equal) real root; if $\Delta < 0$: no real roots (two complex roots).

18. For $ax^{2}+bx+c=0$ with roots $\alpha$ and $\beta$, what are the sum and product of the roots (Vieta's formulas)?

    $\alpha + \beta = -\dfrac{b}{a}$ and $\alpha\beta = \dfrac{c}{a}$.

19. How can you write a quadratic equation given that its roots are $\alpha$ and $\beta$?

    $x^{2} - (\alpha + \beta)x + \alpha\beta = 0$ (with leading coefficient 1).

20. What are the four main methods for solving a quadratic equation?

    Factorising, completing the square, using the quadratic formula, and reading roots from a graph (where the curve crosses the $x$-axis).

21. What is a 'disguised quadratic', and how is it solved?

    An equation that becomes quadratic after a substitution. You substitute $u$ for a repeated expression so it takes the form $au^{2}+bu+c=0$, solve for $u$, then back-substitute to find the original variable.

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