# FPM Polynomial and Curve Sketching (Chapter 4)

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1. (2011/june/Paper02/q4)
A curve has equation $y=x^{3}+2 x^{2}-11 x-m$, where $m$ is a positive integer. The curve crosses the $x$-axis at the point with coordinates $(-4,0)$.

(a) Show that $m=12$ (2 marks)

(b) Factorise $x^{3}+2 x^{2}-11 x-12$ completely. (3 marks)
The curve also crosses the $x$-axis at two other points.

(c) Write down the $x$-coordinate of each of these points. (1 marks

2. (2012/june/paper01/q8)
$$\mathrm{f}(x)=a x^{3}+b x^{2}+c x+d, \text { where } a, b, c \text { and } d \text { are integers. }$$ Given that $\mathrm{f}(0)=6$

(a) show that $d=6$ (1 mark)
When $\mathrm{f}(x)$ is divided by $(x-1)$ the remainder is $-6$
When $\mathrm{f}(x)$ is divided by $(x+1)$ the remainder is 12

(b) Find the value of $b.$ (4 marks)
Given also that $(x-3)$ is a factor of $f(x)$

(c) find the value of $a$ and the value of $c$, (6 marks)

(d) express $\mathrm{f}(x)$ as a product of linear factors. (3 marks)

3. (2013/june/paper02/q6)
$$p(x)=2 x^{3}+13 x^{2}-17 x-70$$

(a) Show that $\mathrm{p}(-2)=0$ (2 marks)

(b) Solve the equation $p(x)=0$ (4 marks)

4. (2014/june/paper02/q9)
$$\mathrm{f}(x)=x^{3}+5 x^{2}+p x-q \quad p, q \in \mathbb{Z}$$ Given that $(x+2)$ and $(x-1)$ are factors of $\mathrm{f}(x)$

(a) form a pair of simultaneous equations in $p$ and $q$, (2 marks)

(b) show that $p=2$ and find the value of $q$, (3 marks)

(c) factorise $\mathrm{f}(x)$ completely. (1 mark)

(d) Sketch the curve with equation $y=\mathrm{f}(x)$ showing the coordinates of the points where the curve crosses the $x$-axis. (2 marks)
The curve with equation $y=x^{3}+2 x^{2}+4 x$ meets the curve with equation $y=\mathrm{f}(x)$ at two points $A$ and $B .$ The $x$-coordinate of $A$ is $-\frac{4}{3}$ and the $x$-coordinate of $B$ is 2

(e) Use algebraic integration to find, to 3 significant figures, the area of the finite region. bounded by the two curves. (5 marks)

5. (2015/jan/paper01/q9)
$\mathrm{f}(x)=2 x^{3}+a x^{2}+b x+15 \quad$ where $a$ and $b$ are constants. The remainder when $\mathrm{f}(x)$ is divided by $(x-1)$ is $-12$ The remainder when $\mathrm{f}(x)$ is divided by $(x+1)$ is 48

(a) Find the value of $a$ and the value of $b$. (6 marks)

(b) Show that $\mathrm{f}\left(\frac{1}{2}\right)=0$ (1 marks)

(c) Express $\mathrm{f}(x)$ as a product of linear factors. (4 marks)

(d) Solve the equation $f(x)=0$ (1 mark)

6. (2016/jan/paper02/q10)
$$\mathrm{f}(x)=2 x^{3}-p x^{2}-13 x-q$$ When $\mathrm{f}(x)$ is divided by $(x-2)$ the remainder is $-20$ Given that $(x-3)$ is a factor of $f(x)$

(a) find the value of $p$ and the value of $q$ (7 marks)

(b) Hence use algebra to solve the equation $f(x)=0$ (5 marks)

7. (2016/june/paper01/q1)
$$\mathrm{f}(x)=x^{3}-7 x+6$$

(a) Show that $(x-2)$ is a factor of $\mathrm{f}(x)$ (2 marks)

(b) Hence, or otherwise, factorise $\mathrm{f}(x)$ completely. (3 marks)

8. (2017/jan/paper01/q2)
$$\mathrm{f}(x)=2 x^{3}-3 p x^{2}+x+4 p \quad \text { where } p \text { is an integer. }$$ Given that $(x-4)$ is a factor of $\mathrm{f}(x)$

(a) show that the value of $p$ is 3. (2 marks)
Using this value of $p$,

(b) find the remainder when $\mathrm{f}(x)$ is divided by $(x+2)$ (2 marks)

(c) factorise $\mathrm{f}(x)$ completely (3 marks)

(d) solve the equation $2 x^{3}-3 p x^{2}+x+4 p=0$ (2 marks)

9. (2019/june/paper01/q1)
$$\mathrm{f}(x)=x^{3}+2 x^{2}-5 x-6$$

(a) Factorise $x^{2}-x-2$ (1 mark)

(b) Hence, or otherwise, show that $\left(x^{2}-x-2\right)$ is a factor of $\mathrm{f}(x)$. (3 marks)

10. (2019/juneR $/$ paper02/q1)
$$\mathrm{f}(x)=(x-3)\left[x^{2}+(p-2) x+q\right]$$ Given that $\mathrm{f}(0)=-12$

(a) find the value of $q$ (5 marks)

(b) Find the range of values of $p$ for which the cubic equation $f(x)=0$ has only one real root. (5 marks)

1.(a) Show (b) $(x+4)(x-3)(x+1)$ (c) $-1,3$

2.(a) Show (b) $b=-3$ (c) $a=2, c=-11$ (d) $f(x)=(x-3)(2 x-1)(x+2)$

3.(a) show (b) $x=-2,-7,2 \frac{1}{2}$

4.(a) $\quad 2 p+q=12, p-q=-6$ (b) Show (c) $(x+2)(x-1)(x+4)$ (d) Graph (e) $18.5$

5.(a) $\quad a=3, b=-32$ (b) Show (c) $(2 x-1)(x+5)(x-3)$ (d) $x=\frac{1}{2},-5,3$

6.(a) $\quad p=1, q=6$ (b) $x=3,-\frac{1}{2},-2$

7.(a) Show (b) $(x-2)(x+3)(x-1)$

8.(a) Show (b) $-42$ (c) $(x-4)(x+1)(2x-3)$ (d) $x=4,-1,\frac{3}{2}$

9.(a) $(x-2)(x+1)$ (b) Show

10.(a) $q=4$ (b) $-2 < p < 6$