Test of Mathematics for University Admission (TMUA) Number, Algebra and Surds Flashcards

24 sample cards from the Number, Algebra and Surds deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Describe the 'splitting the middle term' method for factorising $ax^{2}+bx+c$.

   Find two numbers with product $ac$ and sum $b$, split $bx$ into those two terms, then factorise by grouping.

2. Factorise $6x^{2} + 11x + 3$.

   $ac = 18$, numbers $9$ and $2$: $6x^{2}+9x+2x+3 = 3x(2x+3)+1(2x+3) = (3x+1)(2x+3)$.

3. What is the first step when adding or subtracting algebraic fractions with different denominators?

   Find a common denominator (often the product or lowest common multiple of the denominators) and rewrite each fraction over it before combining numerators.

4. Simplify $\dfrac{x^{2}-9}{x^{2}+4x+3}$.

   Factorise: $\dfrac{(x-3)(x+3)}{(x+1)(x+3)} = \dfrac{x-3}{x+1}$ (for $x\neq -3$).

5. How do you divide one algebraic fraction by another?

   Multiply by the reciprocal of the divisor: $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{ad}{bc}$.

6. What form of partial fractions is used for $\dfrac{px+q}{(ax+b)(cx+d)}$ with distinct linear factors?

   $\dfrac{A}{ax+b} + \dfrac{B}{cx+d}$, where $A$ and $B$ are constants to be determined.

7. What partial-fraction form is required when there is a repeated linear factor $(ax+b)^{2}$?

   $\dfrac{A}{ax+b} + \dfrac{B}{(ax+b)^{2}}$ — one term for each power up to the repeat.

8. What numerator form is needed for an irreducible quadratic factor $(x^{2}+c)$ in partial fractions?

   A linear numerator: $\dfrac{Bx+C}{x^{2}+c}$.

9. Express $\dfrac{5x+1}{(x+1)(x-2)}$ in partial fractions.

   $\dfrac{A}{x+1}+\dfrac{B}{x-2}$ gives $A=-\dfrac{4}{3}$, $B=\dfrac{11}{3}$, so $\dfrac{-4}{3(x+1)} + \dfrac{11}{3(x-2)}$.

10. What must you do before applying partial fractions if the fraction is improper (numerator degree $\geq$ denominator degree)?

    First perform polynomial (long) division to write it as a polynomial plus a proper fraction, then decompose the proper part.

11. Write $x^{2}+6x+5$ in completed-square form.

    $x^{2}+6x+5 = (x+3)^{2} - 9 + 5 = (x+3)^{2} - 4$.

12. For $x^{2}+bx+c$, what is the general completed-square form?

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

13. How does completing the square give the minimum point of $y=(x+p)^{2}+q$?

    The minimum value is $q$, attained at $x=-p$; the vertex is $(-p,\,q)$.

14. Complete the square for $2x^{2}+8x+3$ (note the leading coefficient).

    Factor out $2$: $2(x^{2}+4x)+3 = 2\big((x+2)^{2}-4\big)+3 = 2(x+2)^{2} - 5$.

15. Derive the form that completing the square gives for the roots of $ax^{2}+bx+c=0$.

    It yields the quadratic formula $x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}$.

16. In polynomial long division, what are the dividend, divisor, quotient and remainder related by?

    $\text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder}$, with the remainder of lower degree than the divisor.

17. What is the degree of the product of two polynomials of degrees $m$ and $n$?

    $m+n$ (degrees add when polynomials are multiplied).

18. State the Remainder Theorem.

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

19. State the Factor Theorem.

    $(x-a)$ is a factor of polynomial $f(x)$ if and only if $f(a)=0$.

20. By the Remainder Theorem, what is the remainder when $f(x)=x^{3}-2x+4$ is divided by $(x-2)$?

    $f(2) = 8 - 4 + 4 = 8$, so the remainder is $8$.

21. What is the remainder when $f(x)$ is divided by $(ax-b)$, in terms of $f$?

    $f\!\left(\dfrac{b}{a}\right)$ — the remainder equals $f$ evaluated at the root of $ax-b=0$.

22. For a quadratic $ax^{2}+bx+c=0$ with roots $\alpha,\beta$, state the sum and product of the roots.

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

23. For a cubic $ax^{3}+bx^{2}+cx+d=0$ with roots $\alpha,\beta,\gamma$, give the three symmetric relations.

    $\alpha+\beta+\gamma = -\dfrac{b}{a}$, $\;\alpha\beta+\beta\gamma+\gamma\alpha = \dfrac{c}{a}$, $\;\alpha\beta\gamma = -\dfrac{d}{a}$.

24. How does the discriminant $b^{2}-4ac$ classify the roots of a real quadratic?

    If $b^{2}-4ac>0$: two distinct real roots; if $=0$: one repeated real root; if $<0$: no real roots (a complex conjugate pair).

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