TMUA · Paper 1: Applications of Mathematical Knowledge

Polynomials and the factor theorem

5 practice questions, each with the answer and an explanation.

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All 5 questions

What is the remainder when $x^{3} - 2 x + 5$ is divided by $(x - 2)$ ?
$(x - 2)$ is a factor of $x^{3} + a x^{2} + x - 6$ . What is the value of $a$ ?
The cubic $x^{3} - 3 x^{2} - x + 3 = 0$ has three real roots. What is the sum of the squares of the roots?
How many real roots does the equation $x^{3} - 3 x + 1 = 0$ have?
When a polynomial $f (x)$ is divided by $(x - 1) (x - 2)$ the remainder is $p x + q$ . Given $f (1) = 3$ and $f (2) = 5$ , what is $p x + q$ ?

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