Simplifying and manipulating expressions

Here is a sketch of y = x² + bx + c The graph passes through (– 1, 0) and (5, 0)

Work out the values of b and c.

Solve x³ + 10x² + 8x = 0 Where appropriate give your answers in surd form. Simplify your answers where possible.

(x – 3)(x + 4)(x + p) ≡ x³ + qx² + rx + 24

Work out the values of p, q and r.

k is a positive integer.

Prove that 4 k (k + 8) – (4 k + 9)(k – 1) is a multiple of 9

Factorise fully (x – 1)⁹ + (x – 1)⁸(x + 4)

Factorise fully 8y² – 50

Factorise x² – y²

Solve by factorising 2x³ – 9x² – 5x = 0

Expand and simplify fully (3 x – 1)(4 – x)(2 x + 5)

Factorise fully 6x²y + 21xy

X_n and Y_n are the nth terms of two sequences.

Prove that every term of the sequence with nth term Y_n – X_n is a multiple of 3

Expand and simplify fully (2x³ – 9)(3x² + 4) + x(x – 4)²

(3x a)(5x 4) 15x² 2x b

Work out the values of a and b.

Factorise fully 12w + 18w²

Expand and simplify fully (3 x + 4)(2 x – 3)(5 x – 2)

f(x) = 2x³ + 11x² + 12x – 9

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

Show that f(x) = 0 has exactly two solutions.

y = x(2x⁴ – 7x³)

Work out an expression for the rate of change of y with respect to x.

Expand and simplify fully (2 x – 5)(3 x – 4)(x + 2)

Expand and simplify 5(2 x – 1) + 4(11 – x) Give your answer in the form a (bx + c) where a, b and c are integers greater than 1

Expand and simplify fully (5x + 3y²)(4x – y²)