# 1.1 Remainder and Factor Theorems

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Which of the following are polynomials? Write down the order of those which are.

 Question 1(a) $3x^{2}+\frac{7}{x}$ Question 1(b) $2x^{3}+5$ Question 1(c) $2\sin{x}-4\cos{x}$ Question 1(d) $\left(x+3\right)\left(x^{2}-7x^{3}+1\right)$

Evaluate the following polynomial sums and products:

 Question 2(a) $\left(2x^{3}-3x+1\right)+\left(2x^{3}+4x-4\right)$ Question 2(b) $\left(2x^{2}+3x\right)\times\left(x^{3}-1\right)$ Question 2(c) $\left(x^{3}+2x\right)-\left(x^{2}-x\right)$ Question 2(d) $\left(x^{4}+x^{2}+1\right)\times\left(x^{3}-x\right)$

Find the quotient and remainder in each of the following division problems.

 Question 3(a) $\left(x^{3}+2x^{2}+x-3\right)\div\left(x-4\right)$ Question 3(b) $\left(2x^{2}+3x\right)\div\left(x^{3}-1\right)$ Question 3(c) $\left(x^{3}+2x\right)\div\left(x^{2}-x\right)$ Question 3(d) $\left(x^{4}+x^{2}+1\right)\div\left(x^{3}-x\right)$

Find the remainder in each of the following divisions.

 Question 4(a) $\left(x^{2}+3x-2\right)\div\left(x-5\right)$ Question 4(b) $\left(3x^{3}+x^{2}+x-1\right)\div\left(x-2\right)$ Question 4(c) $\left(x^{3}+7x^{2}-2x-1\right)\div\left(x+1\right)$ Question 4(d) $\left(x^{4}+2\right)\div\left(x+2\right)$

 Question 5 The remainder when $x^{3}-3x^{2}+ax-1$ is divided by $x-2$ is $5$. Find the value of $a$. Question 6 The remainder when $ax^{3}+3x^{2}+3x+5$ is divided by $x+3$ is $8$. Find the value of $a$. Question 7 When $ax^{3}+3x^{2}+3x+5$ is divided by $ax^{3}+3x^{2}+3x+5$ the remainder is $3$, and when it is divided by $ax^{3}+3x^{2}+3x+5$ the remainder is $-1$. Find $a$ and $b$. Question 8 Question 9 Question 10 Question 11 Question 12 Question 13

Factorise the following:

 Question 14(a) $x^{3}-2x^{2}-x+2$ Question 14(b) $x^{3}+3x^{2}-4x-12$ Question 14(c) $x^{3}-2x^{2}-5x+6$ Question 14(d) $2x^{3}+11x^{2}+17x+6$ Question 14(e) $2x^{3}-9x^{2}+7x+6$ Question 14(f) $x^{4}-x^{2}+4x-4$ Question 14(g) $x^{3}-x^{2}-x+1$ Question 14(h) $x^{3}+5x^{2}+8x+4$

Solve the following equations:

 Question 15(a) $x^{3}+4x^{2}+x-6=0$ Question 15(b) $x^{3}-4x^{2}+x+6=0$ Question 15(c) $2x^{3}+5x^{2}+x-2=0$ Question 15(d) $6x^{3}-11x^{2}+6x-1=0$

These questions are from S.C. Solomon's Advanced Level Mathematics.