Pure mathematics
Algebra & Functions: Advanced Techniques
Year 12 · Year 13
- ✓By the end of this lesson students will be able to manipulate and simplify algebraic expressions involving indices and surds.
- ✓By the end of this lesson students will be able to solve quadratic equations and use the discriminant to determine the nature of roots.
- ✓By the end of this lesson students will be able to apply the Factor and Remainder Theorems to factorise and analyse polynomial functions.
- ✓By the end of this lesson students will be able to decompose rational expressions into partial fractions.
- ✓By the end of this lesson students will be able to understand and apply the modulus function and describe transformations of graphs.
Key concepts
Indices (or powers) provide a shorthand notation for repeated multiplication. The laws of indices govern how to manipulate expressions involving powers. These include rules for multiplication, division, raising a power to another power, zero index, negative indices, and fractional indices.
A surd is an irrational number that can be expressed as a root of an integer, such as √2 or ³√5. Surds cannot be expressed as a simple fraction. Simplifying surds involves extracting square factors from under the root sign. Rationalising the denominator involves eliminating surds from the denominator of a fraction by multiplying by an appropriate expression (often the conjugate).
A quadratic equation is a polynomial equation of the second degree. It can be solved by factorising, completing the square, or using the quadratic formula. The solutions are also known as roots.
The quadratic formula provides a direct method to find the roots of any quadratic equation in the form ax^2 + bx + c = 0.
The discriminant, Δ, is the part of the quadratic formula under the square root sign. Its value determines the nature of the roots of a quadratic equation.
A polynomial function is a function of the form P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n, a_{n-1}, ..., a_0 are constants and n is a non-negative integer. The degree of the polynomial is the highest power of x.
The Factor Theorem states that if P(x) is a polynomial, then (x - a) is a factor of P(x) if and only if P(a) = 0. This is a powerful tool for factorising polynomials.
The Remainder Theorem states that when a polynomial P(x) is divided by a linear expression (x - a), the remainder is P(a).
Partial fractions is a technique used to decompose a complex rational expression (a fraction where the numerator and denominator are polynomials) into a sum of simpler fractions. This is particularly useful in integration and for solving differential equations. Different forms are used depending on the nature of the factors in the denominator (distinct linear, repeated linear, irreducible quadratic).
The modulus function, denoted |x|, gives the absolute value of a number x. It is defined as x if x ≥ 0, and -x if x < 0. Geometrically, |x| represents the distance of x from zero on the number line.
Graph transformations involve altering the position, size, or orientation of a graph. Key transformations include translations (shifting), stretches (scaling), and reflections (flipping). These are applied to a base function y = f(x).
Key facts to remember
- 1Laws of indices: a^m * 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^(m/n) = (ⁿ√a)^m.
- 2For surds, √ab = √a * √b and √(a/b) = √a / √b. Always simplify surds by extracting square factors.
- 3To rationalise a denominator of the form (a ± √b), multiply by its conjugate (a ∓ √b).
- 4The discriminant Δ = b^2 - 4ac determines the nature of roots for ax^2 + bx + c = 0: Δ > 0 (two distinct real roots), Δ = 0 (one repeated real root), Δ < 0 (no real roots, two complex conjugate roots).
- 5Factor Theorem: (x - a) is a factor of P(x) if and only if P(a) = 0.
- 6Remainder Theorem: The remainder when P(x) is divided by (x - a) is P(a).
- 7Partial fraction decomposition forms depend on the denominator's factors: A/(ax+b) for distinct linear, A/(ax+b) + B/(ax+b)^2 for repeated linear, (Ax+B)/(ax^2+bx+c) for irreducible quadratic.
- 8The modulus function |x| means the positive value of x. For equations |f(x)| = k, solve f(x) = k and f(x) = -k.
- 9Transformations: y = f(x) + a (vertical translation by a), y = f(x + a) (horizontal translation by -a), y = af(x) (vertical stretch by factor a), y = f(ax) (horizontal stretch by factor 1/a), y = -f(x) (reflection in x-axis), y = f(-x) (reflection in y-axis).
Worked examples
Example 1
Simplify the expression: (27x^6)^(2/3) / (9x^-2)
Answer
x^6
Remember to apply the power to both the coefficient and the variable term in the bracket.
Example 2
Rationalise the denominator of (3 + √2) / (5 - √2).
Answer
(17 + 8√2) / 23
The difference of two squares identity is crucial for rationalising binomial surd denominators.
Example 3
Find the range of values for k such that the quadratic equation x^2 + kx + 4 = 0 has real roots.
Answer
k ≤ -4 or k ≥ 4
Real roots include both distinct real roots (Δ > 0) and repeated real roots (Δ = 0).
Example 4
Show that (x - 2) is a factor of f(x) = x^3 - 3x^2 + 4, and hence factorise f(x) completely.
Answer
f(x) = (x - 2)(x - 2)(x + 1) = (x - 2)^2 (x + 1)
Always check for repeated factors after finding the first one.
Example 5
Decompose (5x - 1) / ((x + 1)(x - 2)) into partial fractions.
Answer
2 / (x + 1) + 3 / (x - 2)
Substituting values of x that make the denominators zero is usually the quickest method for distinct linear factors.
Example 6
Solve the equation |2x - 3| = 5.
Answer
x = 4 or x = -1
Always consider both positive and negative cases when solving modulus equations.
Example 7
Describe the sequence of transformations that map the graph of y = x^2 to the graph of y = 2(x - 1)^2 + 3.
Answer
1. Horizontal translation by 1 unit in the positive x-direction (right). 2. Vertical stretch by a factor of 2 parallel to the y-axis. 3. Vertical translation by 3 units in the positive y-direction (upwards).
For transformations of the form y = af(bx+c)+d, remember that horizontal transformations (bx+c) often involve factoring out 'b' first, e.g., f(b(x+c/b)).
Common mistakes
- ✗Incorrectly applying index laws, especially with negative or fractional indices (e.g., confusing a^-n with -a^n or (a^m)^n with a^m * a^n).
- ✗Algebraic errors when solving quadratics (e.g., sign errors in the quadratic formula) or when performing algebraic long division for polynomials.
- ✗Misinterpreting the discriminant conditions, particularly confusing 'real roots' (Δ ≥ 0) with 'distinct real roots' (Δ > 0).
- ✗Setting up incorrect forms for partial fractions, especially for repeated linear factors or irreducible quadratic factors.
- ✗Errors with modulus equations/inequalities, such as forgetting to consider both positive and negative cases, or incorrectly squaring both sides.
- ✗Confusing the direction or type of graph transformation (e.g., mixing up horizontal and vertical translations or stretches, or the order of transformations).
Exam tips
- ★For indices and surds, show all steps of simplification to avoid calculation errors and to gain method marks.
- ★When using the discriminant, clearly state the condition (e.g., Δ ≥ 0 for real roots) before substituting values and solving the inequality.
- ★For polynomial factorisation, if you are given a factor or root, use it to simplify the polynomial before attempting to find other factors.
- ★When decomposing partial fractions, check your answer by recombining the partial fractions to see if you get the original expression.
- ★For graph transformations, apply stretches/reflections before translations. If multiple transformations are involved, describe them in a clear, logical sequence.
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