📄 Cheat Sheets — Pure Mathematics 2 & 3

Key facts, formulas and common mistakes for 1 of 1 chapters

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8 key facts·6 formulas·6 common mistakes·18 definitions·6 exam tips

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Definitions

modulus: The modulus of a number is the magnitude of the number without a sign attached.
absolute value: The modulus of a number is also called the absolute value.
polynomial: A polynomial is an expression of the form a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0, where x is a variable, n is a non-negative integer, the coefficients a_0, a_1, ..., a_n are constants, a_n is called the leading coefficient and a_n ≠ 0, and a_0 is called the constant term.
leading coefficient: In a polynomial a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0, a_n is called the leading coefficient and a_n ≠ 0.
constant term: In a polynomial a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0, a_0 is called the constant term.
degree of the polynomial: The highest power of x in the polynomial is called the degree of the polynomial.
dividend: In polynomial division, the dividend is the polynomial being divided.
divisor: In polynomial division, the divisor is the polynomial by which another polynomial is divided.
quotient: In polynomial division, the quotient is the result of the division, excluding any remainder.
remainder: In polynomial division, the remainder is the polynomial left over after the division process, which has a degree less than the divisor.
factor: If a polynomial P(x) divides exactly by a linear factor (x - c) to give the polynomial Q(x), then (x - c) is a factor of P(x).
factor theorem: If for a polynomial P(x), P(c) = 0, then (x - c) is a factor of P(x).
remainder theorem: If a polynomial P(x) is divided by (x - c), the remainder is P(c).
x-intercept: The x-intercept is the point where the graph meets the x-axis.
y-intercept: The y-intercept is the point where the graph meets the y-axis.
vertex: The vertex of a modulus function graph of the form y = |ax + b| is the point where the graph changes direction, forming a 'V' shape.
intersection: An intersection point is a point where two or more graphs meet.
inequality: An inequality is a mathematical statement that compares two expressions using an inequality symbol (e.g., <, >, ≤, ≥).

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Formulas

Ch 01

Modulus of x

∣ x ∣ = {x - x if x \geq 0 if x < 0

Ch 01

Modulus equation property 1

∣ x ∣ = a ⟺ x = a or x = - a

Ch 01

Modulus equation property 2

∣ a x + b ∣ = ∣ c x + d ∣ ⟺ (a x + b)^{2} = (c x + d)^{2}

Ch 01

Modulus inequality property 1

∣ x ∣ < a ⟺ - a < x < a

Ch 01

Modulus inequality property 2

∣ x ∣ > a ⟺ x < - a or x > a

Ch 01

Division algorithm for polynomials

P (x) = D (x) Q (x) + R (x)

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Key Facts

The modulus function |x| represents the magnitude of x, meaning |x| = x if x ≥ 0 and |x| = -x if x < 0.Ch 01
To solve |x| = a, consider both x = a and x = -a.Ch 01
To solve |ax + b| = |cx + d|, square both sides: (ax + b)² = (cx + d)².Ch 01
For modulus inequalities, |x| < a means -a < x < a, and |x| > a means x < -a or x > a.Ch 01
The division algorithm for polynomials states P(x) = D(x)Q(x) + R(x), where P(x) is the dividend, D(x) is the divisor, Q(x) is the quotient, and R(x) is the remainder.Ch 01
The Factor Theorem states that if P(c) = 0 for a polynomial P(x), then (x - c) is a factor of P(x).Ch 01
The Remainder Theorem states that when a polynomial P(x) is divided by (x - c), the remainder is P(c).Ch 01
Graphs of y = |ax + b| form a 'V' shape, with the vertex at the x-intercept of ax + b = 0.Ch 01

⚠️

Common Mistakes

✗Don't forget to check for extraneous roots when solving modulus equations by squaring both sides, especially for forms like |ax + b| = cx + d.Ch 01
✗Don't confuse the conditions for modulus inequalities: |x| < a means 'between' (-a < x < a), while |x| > a means 'outside' (x < -a or x > a).Ch 01
✗Remember to include zero coefficients for any missing terms (e.g., 0x²) when performing polynomial long division to maintain correct alignment.Ch 01
✗Don't confuse the Factor Theorem and the Remainder Theorem; the Factor Theorem is a special case of the Remainder Theorem where the remainder is zero.Ch 01
✗Remember to reverse the inequality sign when multiplying or dividing by a negative number during algebraic manipulation.Ch 01
✗Don't fail to completely factorise polynomials; always check if quadratic factors can be further broken down.Ch 01

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Exam Tips

→For modulus equations and inequalities, always consider both algebraic methods (squaring, definition) and graphical interpretation to verify solutions.Ch 01
→When sketching modulus graphs, clearly label the vertex (x-intercept of the original function) and any intercepts with the axes.Ch 01
→Show all steps clearly for polynomial long division, especially when dealing with missing terms, to avoid calculation errors and gain method marks.Ch 01
→When applying the Factor or Remainder Theorem, explicitly state P(c) and its value to demonstrate understanding.Ch 01
→For cubic and quartic equations, use the Factor Theorem to find initial factors, then use algebraic division to reduce the polynomial to a quadratic, which can then be factorised or solved.Ch 01
→Always present solutions to inequalities using correct notation (e.g., -a < x < a or x < -a, x > a).Ch 01

📄 Cheat Sheets — Pure Mathematics 2 & 3

Pure Mathematics 2 & 3 — Cheat Sheets

Definitions

Formulas

Key Facts

Common Mistakes

Exam Tips

📄 Cheat Sheets — Pure Mathematics 2 & 3

Pure Mathematics 2 & 3 — Cheat Sheets

Definitions

Formulas

Key Facts

Common Mistakes

Exam Tips