Sec 3 A-Math Polynomials & Partial Fractions topical questions

Divide x² + 7x + 12 by (x + 3), stating the quotient.

Find the remainder when P(x) = 2x³ − x² + 4x − 5 is divided by (x − 1), and state what the answer tells you about the factorisation of P(x).

Find the remainder when x³ + 2x² − 5x + 1 is divided by (x + 1).

Factorise x³ − 8 completely.

Given that (x − 1) is a factor of x³ + kx² + 2x − 3, find the value of k.

Solve the equation x³ + 2x² − 5x − 6 = 0.

Express (7x − 6)/((x − 2)(x + 1)) in partial fractions.

Express (2x³ − 3x + 4)/(x − 1) as a polynomial plus a partial fraction.

Decompose (4x − 5)/((x − 1)(x − 2)²) into partial fractions.

The polynomial P(x) = x³ + ax² + bx − 3 leaves remainder 5 when divided by (x − 1) and remainder −9 when divided by (x + 1). Find a and b.

The polynomial 2x³ + px² − 7x + q is divisible by both (x − 1) and (x + 2). Find the values of p and q.

Solve the equation x³ + 3x² + 3x − 7 = 0.

1. (a)Show that (x + 1)³ = x³ + 3x² + 3x + 1.[2]
2. (b)Rewrite the equation using part (a) in the form (x + 1)³ = 8.[2]
3. (c)Hence solve the equation, explaining why there is only one real solution.[2]

Express (5x² + 6x + 5)/((x + 1)(x² + 1)) in partial fractions.

The polynomial f(x) = x³ − 6x² + ax + b is divisible by both (x − 1) and (x + 2).

1. (a)Form two equations in a and b.[2]
2. (b)Find a and b.[2]
3. (c)Hence solve f(x) = 0 completely.[2]

Express (3x² + 14x + 15)/((x + 1)(x + 2)²) in partial fractions.