Polynomials and Partial Fractions
GCE Further Mathematics

1. (a) Given that f(x) = 2x³ + 3x² - 11x - 6, show that (x - 2) is a factor of f(x). [2 marks] (b) Hence, factorize f(x) completely. [4 marks] (c) Express (7x² - 11x - 6) / [(2x + 1)(x - 2)(x + 3)] …

2. (a) The polynomial p(x) = x⁴ - 3x³ + ax² + bx - 10 has (x - 1) and (x + 2) as factors. Find the values of the constants a and b. [6 marks] (b) Using the values of a and b found in (a), factorize p…

If $(x - 2)$ is a factor of $f(x) = x^3 + kx^2 - 4x + 4$, find the value of $k$.

Express $\dfrac{5x - 1}{(x+1)(x-2)}$ in partial fractions.

When the polynomial $f(x) = x^3 - 2x^2 + ax + b$ is divided by $(x - 1)$, the remainder is 4, and when divided by $(x + 1)$, the remainder is -2. Find the values of $a$ and $b$.

If f(x) = 2x³ - 3x² + x - 5, find f(2).

Express $\dfrac{3x^2 + x + 2}{(x^2 + 1)(x - 1)}$ in partial fractions.

The polynomial $g(x) = x^3 + px^2 - 7x + q$ has $(x - 2)$ as a factor and leaves a remainder of 24 when divided by $(x + 3)$. Find $p + q$.

Which of the following is the correct partial fraction decomposition of $\dfrac{7}{(x-1)(x+6)}$?

Find the remainder when $f(x) = 2x^4 - 3x^3 + x^2 - 5x + 2$ is divided by $(x + 1)$.

Which of the following is a factor of the polynomial $p(x) = x^3 - 6x^2 + 11x - 6$?