Proof by deduction

Given that

• f(x) = 2x³ + 3x² – 16x + 16

• f(–4) = 0

write f(x) in the form (x + a) Q (x) where a is a constant and Q(x) is a quadratic expression.

Hence prove that –4 is the only real root of the equation f(x) = 0 (Solutions relying on calculator technology are not acceptable.)

A student was asked to prove that the square of any number can be expressed in the form 5n or 5n ± 1 where n The start of the student’s proof is shown in the box below.

Identify and correct an algebraic error in the box above.

Show the calculations and statements that are required to complete the proof.

Show thatk² – 4k + 5 is positive for all real values of k.

A student was asked to prove by contradiction that “There are no positive integers x and y such that 3x 2y2x 5y 28 ”

The start of the student’s proof is shown below.

Show the calculations and statements needed to complete the proof.

Prove, using algebra, that

is odd for all n

Prove, using algebra, that

is even for all n .

Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3

Prove that for all n , n² + 2 is not divisible by 4

“Given x , the value of 3x – 28 is greater than or equal to the value of (x – 9).” State, giving a reason, if the above statement is always true, sometimes true or never true.