Index and logarithm laws, factorisation techniques, solving polynomial equations, and the relationship between exponential and logarithmic functions

What this dot point is asking

QCAA wants Year 11 students to be fluent with index and logarithm laws, factorise polynomials, and solve linear, quadratic, polynomial, exponential and logarithmic equations.

Index laws

$a^m \cdot a^n = a^{m+n}$. $a^m / a^n = a^{m-n}$. $(a^m)^n = a^{mn}$.

$a^0 = 1$. $a^{-n} = 1/a^n$. $a^{1/n} = \sqrt[n]{a}$.

$(ab)^n = a^n b^n$. $(a/b)^n = a^n / b^n$.

Logarithm laws

$\log_a(mn) = \log_a m + \log_a n$.

$\log_a(m/n) = \log_a m - \log_a n$.

$\log_a(m^n) = n \log_a m$.

$\log_a(1) = 0$. $\log_a(a) = 1$.

Change of base: $\log_a x = \log_b x / \log_b a$.

Inverse: $\log_a(a^x) = x$, $a^{\log_a x} = x$.

Factorisation

Common factor. $6x^3 - 9x^2 = 3x^2(2x - 3)$.

Grouping. $x^3 + 2x^2 - x - 2 = x^2(x + 2) - (x + 2) = (x + 2)(x^2 - 1)$.

Quadratic. $x^2 + 5x + 6 = (x + 2)(x + 3)$.

Quadratic formula. $x = (-b \pm \sqrt{b^2 - 4ac})/(2a)$.

Difference of squares. $a^2 - b^2 = (a-b)(a+b)$.

Sum/difference of cubes. $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$. $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$.

Solving equations

Linear. Single-step manipulation.

Quadratic. Factor first, use null factor law. Or quadratic formula.

Polynomial. Factor where possible. Find rational roots first; polynomial division for higher degree.

Exponential. Bring to common base, equate exponents. Otherwise take logs.

Logarithmic. Combine logs using laws; convert to exponential form. Always check domain.

Common errors

Sign on negative exponent. $a^{-n} = 1/a^n$, not $-a^n$.

Log of negative. Undefined; always check domain.

Wrong factorisation. $a^2 - b^2$ factors; $a^2 + b^2$ does not (over reals).

Forgetting both quadratic roots. Report both.

Missing log domain check. Solutions that produce negative log arguments must be rejected.

In one sentence

Unit 1 algebra establishes fluency with index laws, logarithm laws (including change of base), polynomial factorisation (common factor, grouping, quadratic, difference of squares, sum/difference of cubes) and the solution of linear, quadratic, polynomial, exponential and logarithmic equations; the mandatory domain check on logarithmic solutions is the most-tested detail.