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1. Given that $$2x^3-7x^2+13+c$$ can be written as $(2x-3)(x^2+ax+b)$, find $a$, $b$ and $c$
2. Given that $2x-1$ is a factor of the denominator, fully simplify $$\dfrac{4x^2+16x}{2x^3+3x^2-18x+8}$$
3. Given that $2x+1$ is a factor of the numerator, fully simplify $$\dfrac{4x^3-7x-3}{2x^2+3x+1}$$
4. Given that $5x-2$ is a factor of the denominator, fully simplify $$\dfrac{15x^2-6x}{15x^3+19x^2-4}$$
5. Given that $3x+1$ is a factor of the denominator, fully simplify $$\dfrac{9x^3+21x^2+6x}{9x^3+18x^2-x-2}$$
6. When $$3x^3+2x^2-7x+d$$ is divided by $3x-1$, the remainder is $-1$. Find $d$.
7. Write $$\dfrac{2x^3-x^2+2x-2}{2x-1}$$ in the form $x^2+a+\dfrac{b}{2x-1}$
8. Given that $(2x+1)$ is a factor of $$6x^3-19x^2+9x+10$$ fully factorise it.
9. Given that $2x-3$ is a factor of the denominator, fully simplify $$\dfrac{2x^2+x-6}{4x^3-13x+6}$$
10. Determine whether any of: $(x-2)$, $(x-4)$ or $(x-6)$ are factors of $$x^3-3x^2+6x-40$$
11. Use polynomial division to simplify $$\dfrac{x^3+4x^2+7x+6}{x+2}$$
12. Given that $x-a$ is a factor of $$x^3+x^2-x+a$$ find the possible values of $a$.
13. The remainder when $$ax^3 - 32x^2 - 10x + 12$$ is divided by $x-2$ is the same as when it is divided by $2x+3$. Find $a$.
14. One solution to the equation $$3x^3-2x^2-12x+8=0$$ is $x=2$. Find the other solutions.
15. Given that $x = -3$ is a solution of the equation $$2x^3+3x^2-8x+a$$ find two other solutions to the equation.
16. The graph of $$y = 2x^3-7x^2-2x+1$$ crosses the $x$ axis at $x=-0.5$. Find the exact values of any other intersections with the $x$ axis.
17. The graph of $$y = 6x^3-7x^2-x+2$$ crosses the $x$ axis at $x=1$. Find the exact values of any other intersections with the $x$ axis.
18. Show that the graph of $$y=2x^3+x^2+8x+4$$ crosses the $x$ axis only once, at $x = -0.5$.