Comprehensive assessment covering polynomial basics, division, factor and remainder theorems, Vieta's formulas, graphing, inequalities, and applications.
What is the degree of $P(x) = -2x^5 + 3x^4 - x^2 + 7$?
When $P(x)$ is divided by $x - a$, the remainder is:
If $x - 3$ is a factor of $P(x)$, then:
The leading coefficient of $P(x) = 4x^3 - x^5 + 2$ is:
Dividing $x^3 - 8$ by $x - 2$ gives remainder:
For $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, $\alpha + \beta$ equals:
For $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, $\alpha \beta$ equals:
If $x = 1$ and $x = -2$ are zeros of a cubic $P(x)$, then $P(x)$ has factors:
The end behaviour of $y = -x^4 + 3x$ is:
A root of even multiplicity usually means the graph:
A root of odd multiplicity usually means the graph:
The zeros of $P(x) = x(x - 2)(x + 5)$ are:
If $P(x) = x^3 - 4x^2 + 2x - 1$ has roots $\alpha$, $\beta$, $\gamma$, then $\alpha + \beta + \gamma$ equals:
Which expression is fully factorised over the reals?
To solve a polynomial inequality using a sign diagram, the first key step is to:
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)