Mathematical Proof Methods: Exhaustion and Counter Example

Proof by exhaustion = trying all the options Disproof by counter example = showing that a statement is untrue Factor Theorem = using the equation of a circle in the form (x − a)2 + (y − b)2 = r2 Remainder Theorem = proceeding from given assumptions through a series of logical steps

Quotient = the result of dividing one polynomial by another Remainder = the amount left over when one polynomial is divided by another Factor = a polynomial multiplied by another to get a product Divisor = a polynomial that divides another polynomial

The angle in a semicircle is a right angle = a characteristic of a circle's centre The perpendicular from the centre to a chord bisects the chord = a property of a circle's radius The perpendicularity of radius and tangent = a characteristic of a circle's circumference The equation of a circle in the form (x − a)2 + (y − b)2 = r2 = a property of a circle's diameter

<p>Coordinate geometry = proving that a statement is untrue Algebraic division = finding the equation of a circle Factor Theorem = dividing a polynomial by (ax + b) Remainder Theorem = determining the remainder of a polynomial division</p> Signup and view all the answers

<p>Cubic expressions = x3 + 3x2 − 4 and 6x3 + 11x2 − x − 6 Linear equations = x + 2 = 5 Quadratic equations = x2 + 4x + 4 = 0 Polynomial long division = dividing a polynomial by (ax + b)</p> Signup and view all the answers

<p>Arithmetic sequences and series = Sequences and series Laws of logarithms = Exponentials and logarithms Binomial expansion = Algebra Trigonometric equations = Trigonometry</p> Signup and view all the answers

<p>tanθ = sinθ/cosθ = Trigonometry Σ notation = Sequences and series $a^x$ = Exponentials and logarithms $n!$ = Binomial expansion</p> Signup and view all the answers

<p>2sin(x) + 3 = 4 = Trigonometric equations $ax = b$ = Exponentials and logarithms $x^n + 1 = f(x^{n-1})$ = Sequences and series $y = ax$ = Exponentials and logarithms</p> Signup and view all the answers

<p>Sum of a finite arithmetic series = Formula for the nth term Sum to infinity of a convergent geometric series = Formula for the sum of a finite geometric series Binomial expansion of (a + bx)^n = n! and nCr notation Laws of logarithms = Logarithmic identities</p> Signup and view all the answers

<p>y = ax = Exponential graph sin^2(x) + cos^2(x) = 1 = Trigonometric graph Arithmetic sequence = Linear graph Geometric sequence = Exponential graph</p> Signup and view all the answers