Comprehensive assessment covering polynomial basics, division, factor and remainder theorems, Vieta's formulas, graphing, inequalities, and applications.
1. Find the degree and leading coefficient of $P(x)=-2x^5+3x^4-x^2+7$. (2 marks)
2. Use polynomial long division to divide $x^3+4x^2-3x+2$ by $x+2$. State the quotient and remainder. (3 marks)
3. When $P(x)=2x^3+kx^2-5x+3$ is divided by $(x-1)$, the remainder is 4. Find $k$. (2 marks)
4. Fully factorise $x^3-3x^2-4x+12$. (3 marks)
5. For $3x^2-7x+2=0$, find the sum and product of the roots. (2 marks)
6. If $\alpha,\beta,\gamma$ are roots of $x^3-4x^2+2x-1=0$, find $\alpha^2+\beta^2+\gamma^2$. (2 marks)
7. Sketch $y=(x-2)^2(x+1)$, showing all intercepts and end behaviour. (4 marks)
8. Solve $x^3-2x^2-3x\le0$. (3 marks)
9. A rectangular box has a square base. The height is 3 cm more than the side of the base. If the volume is 180 cm³, find the dimensions. (3 marks)
10. $P(x)=x^3+ax^2+bx-6$ has $(x-1)$ and $(x+2)$ as factors. Find $a$ and $b$. (4 marks)