Polynomials

In this question you must show detailed reasoning.

A curve has parametric equations x = t³ + t², y = t² + 2t for all real values of t.

The curve passes through the point P with coordinates (2, 3).

Show that the equation of the tangent to the curve at the point P can be written as 5 y = 4 x + 7

Determine the coordinates of the point where the tangent to the curve at P meets the curve

again.

In this question you must show detailed reasoning.

The cubic polynomial f(x) is defined by f(x) = 2x³ - 3x² - 11x + 6.

Use the factor theorem to show that (2 x - 1) is a factor of f (x).

Express f (x) in fully factorised form.

Hence solve the equation 2 × 8^y - 3 × 4^y - 11 × 2^y + 6 = 0.

The number K is defined by K = n³ + 1, where n is an integer greater than 2.

Given that n³ + 1 ≡ (n + 1)(n² + bn + c), find the constants b and c.

Prove that K has at least two distinct factors other than 1 and K.

In this question you must show detailed reasoning.

The functions f and g are defined for all real values of x by

Write down expressions for

fg (x),

gf (x).

Hence find the values of x for which fg (x) - gf (x) = 24.