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

Define an irrational number and give two examples.

A number that cannot be written as a ratio of two integers; its decimal expansion is non-terminating and non-recurring. Examples: $\pi$ and $\sqrt{2}$.

Is the sum of a rational number and an irrational number rational or irrational? Justify briefly.

Irrational. If $r$ (rational) $+ x$ (irrational) were rational, then $x = (\text{rational}) - r$ would be rational, a contradiction.

Outline the classic proof that $\sqrt{2}$ is irrational.

Assume $\sqrt{2}=\dfrac{p}{q}$ in lowest terms. Then $p^{2}=2q^{2}$, so $p$ is even, $p=2k$, giving $q^{2}=2k^{2}$, so $q$ is even too — contradicting lowest terms. Hence $\sqrt{2}$ is irrational.

Is the product of two irrational numbers always irrational? Give a counterexample.

No. $\sqrt{2}\times\sqrt{2} = 2$, which is rational.

State the difference of two squares factorisation.

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

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

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

Factorise the quadratic $x^{2} + 5x + 6$.

Find two numbers multiplying to $6$ and summing to $5$ ($2$ and $3$): $x^{2}+5x+6 = (x+2)(x+3)$.

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.

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

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

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.

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 $x\neq -3$).

How do you divide one algebraic fraction by another?

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

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.

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.

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

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

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 $\dfrac{-4}{3(x+1)} + \dfrac{11}{3(x-2)}$.

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.

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

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

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

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

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 $(-p,\,q)$.

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

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

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

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

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.

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

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

State the Remainder Theorem.

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

State the Factor Theorem.

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

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 $8$.

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 $ax-b=0$.

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 $\alpha\beta = \dfrac{c}{a}$.

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

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

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).