CIE Polynomial (-2020)

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1. (CIE $\left.0606 / 2018 / \mathrm{w} / 11 / \mathrm{q}^{2}\right)$

$$\begin{aligned}&p(x)=2 x^{3}+5 x^{2}+4 x+a \\&q(x)=4 x^{2}+3 a x+b\end{aligned}$$

Given that $\mathrm{p}(x)$ has a remainder of 2 when divided by $2 x+1$ and that $\mathrm{q}(x)$ is divisible by $x+2$,

(i) find the value of each of the constants $a$ and $b_{i}$

Given that $\mathrm{r}(x)=\mathrm{p}(x)-\mathrm{q}(x)$ and using your values of $a$ and $b$,

(ii) find the exact remainder when $\mathrm{r}(x)$ is divided by $3 x-2$.


2. (CIE $0606 / 2018 / \mathrm{w} / 12 / \mathrm{q} 5)$

The $7^{\text {th }}$ term in the expansion of $(a+b x)^{12}$ in ascending powers of $x$ is $924 x^{6}$. It is given that $a$ and $b$ are positive constants.

(i) Show that $b=\frac{1}{a}.$

The $6^{\text {th }}$ term in the expansion of $(a+b x)^{12}$ in ascending powers of $x$ is $198 x^{5}$.

(ii) Find the value of $a$ and of $b .$


3. (CIE $0606 / 2018 / \mathrm{w} / 12 / \mathrm{q} 9)$

The polynomial $p(x)=a x^{3}+b x^{2}+c x-9 \quad$ is divisible by $x+3$. It is given that $p^{\prime}(0)=36$ and $p^{\prime \prime}(0)=86$.

(i) Find the value of each of the constants $a, b$ and $c$.

(ii) Using your values of $a, b$ and $c$, find the remainder when $p(x)$ is divided by $2 x-1$.$[2]$


4. (CIE $0606 / 2018 / \mathrm{w} / 13 / \mathrm{q} 11)$

The polynomial $\mathrm{p}(x)=a x^{3}+17 x^{2}+b x-8$ is divisible by $2 x-1$ and has a remainder of $-35$ when divided by $x+3$

(i) By finding the value of each of the constants $a$ and $b$, verify that $a=b$.

Using your values of $a$ and $b$,

(ii) find $\mathrm{p}(x)$ in the form $(2 x-1) \mathrm{q}(x)$, where $\mathrm{q}(x)$ is a quadratic expression,

(iii) factorise $\mathrm{p}(x)$ completely,$[1]$

(iv) solve $a \sin ^{3} \theta+17 \sin ^{2} \theta+b \sin \theta-8=0$ for $0^{\circ}<\theta<180^{\circ}$.

5. $(\mathrm{CIE} 0606 / 2019 / \mathrm{m} / 12 / \mathrm{q} 4)$

The polynomial $p(x)=2 x^{3}+a x^{2}+b x-49$, where $a$ and $b$ are constants. When $p^{\prime}(x)$ is divided by $x+3$ there is a remainder of $-24$

(i) Show that $6 a-b=78$.

It is given that $2 x-1$ is a factor of $p(x)$.

(ii) Find the value of $a$ and of $b$.

(iii) Write $p(x)$ in the form $(2 x-1) \mathrm{Q}(x)$, where $\mathrm{Q}(x)$ is a quadratic factor.

(iv) Hence factorise $\mathrm{p}(x)$ completely.


6. (CIE $0606 / 2019 / \mathrm{w} / 21 / \mathrm{q} 7)$

(a) (i) Use the factor theorem to show that $2 x-1$ is a factor of $p(x)$, where $p(x)=4 x^{3}+9 x-5$.

(ii) Write $\mathrm{p}(x)$ as a product of linear and quadratic factors.

(b) (i) Show that $13 \tan x \sec x-4 \sin x-5 \sec ^{2} x=0 \quad$ can be written as $4 \sin ^{3} x+9 \sin x-5=0$

(ii) Using your answers to part (a)(ii) and part (b)(i) solve the equation

$$13 \tan x \sec x-4 \sin x-5 \sec ^{2} x=0 \quad \text { for } 0<x<2 \pi \text { radians. }$$


7. (CIE $0606 / 2019 / \mathrm{s} / 22 / \mathrm{q} 3)$

(i) Given that $x-2$ is a factor of $a x^{3}-12 x^{2}+5 x+6$, use the factor theorem to show that $a=4$. [2]

(ii) Showing all your working, factorise $4 x^{3}-12 x^{2}+5 x+6$ and hence solve $4 x^{3}-12 x^{2}+5 x+6=0$.


8. (CIE $0606 / 2019 / \mathrm{s} / 23 / \mathrm{q} 12)$

Do not use a calculator in this question.

The line $y=4 x-6$ intersects the curve $y=10 x^{3}-19 x^{2}-x$ at the points $A, B$, and $C$. Given that $C$ is the point $(2,2)$, find the coordinates of the midpoint of $A B$.


9. (CIE $0606 / 2019 / \mathrm{w} / 23 / \mathrm{q} 3)$

The first four terms in the expansion of $(1+a x)^{5}(2+b x)$ are $2+32 x+210 x^{2}+c x^{3}$, where $a, b$ and $c$ are integers. Show that $3 a^{2}-16 a+21=0$ and hence find the values of $a, b$ and $c$.


10. (CIE $0606 / 2019 / \mathrm{w} / 23 / \mathrm{q} 8)$

The roots of the equation $x^{3}+a x^{2}+b x+24=0 \quad$ are 2,3 and $p$, where $p$ is an integer.

(i) Find the value of $p$.[1]

(ii) Show that $a=-1$ and find the value of $b$.

Given that a curve has equation $y=x^{3}-x^{2}+b x+24$ find, using your value of $b$

(iii) $\frac{\mathrm{d} y}{\mathrm{dx}}$,

(iv) the integer value of $x$ for which the gradient of the curve is 2 and the corresponding value of $y$.

The coordinates of the point $P$ on the curve are given by the values of $x$ and $y$ found in part (iv).

(v) Find the equation of the tangent to the curve at $P$.$[1]$


11. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 12 / \mathrm{q} 10)$

The polynomial $\mathrm{p}(x)=6 x^{3}+a x^{2}+b x+2$, where $a$ and $b$ are integers, has a factor of $x-2$

(a) Given that $p(1)=-2 p(0)$, find the value of $a$ and of $b$.

(b) Using your values of $a$ and $b$

(i) find the remainder when $p(x)$ is divided by $2 x-1$

(ii) factorise $\mathrm{p}(x)$


12. $(\mathrm{CIE} 0606 / 2020 / \mathrm{m} / 12 / \mathrm{q} 7)$

$\mathrm{p}(x)=a x^{3}+3 x^{2}+b x-12$ has a factor of $2 x+1$. When $\mathrm{p}(x)$ is divided by $x-3$ the remainder is 105 .

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

(b) Using your values of $a$ and $b$, write $\mathrm{p}(x)$ as a product of $2 x+1$ and a quadratic factor.$[2]$

(c) Hence solve $\mathrm{p}(x)=0$.


13. (CIE $0606 / 2020 / \mathrm{s} / 13 / \mathrm{q} 5)$

$p (x) = 6x^3 + ax^2 + 12x + b ,$ where $a$ and $b$ are integers. $p (x)$ has a remainder of 11 when divided by $x - 3$ and a remainder of $- 21$ when divided by $x + 1.$

(a) Given that $p (x) = (x - 2) Q (x) ,$ find $Q (x) ,$ a quadratic factor with numerical coefficients. [6]

(b) Hence solve $p (x) = 0 .$ [2]


14. (CIE $0606 / 2020 / \mathrm{w} / 13 / \mathrm{q} 7)$

The polynomial $\mathrm{p}(x)=a x^{3}+b x^{2}-19 x+4$, where $a$ and $b$ are constants, has a factor $x+4$ and is such that $2 \mathrm{p}(1)=5 \mathrm{p}(0)$.

(a) Show that $\mathrm{p}(x)=(x+4)\left(A x^{2}+B x+C\right)$, where $A, B$ and $C$ are integers to be found.$[6]$

(b) Hence factorise $\mathrm{p}(x)$.$[1]$

(c) Find the remainder when $p^{\prime}(x)$ is divided by $x$.$[1]$


15. (CIE $0606 / 2020 / \mathrm{s} / 21 / \mathrm{q} 3)$

DO NOT USE A CALCULATOR IN THIS QUESTION.

$$p(x)=15 x^{3}+22 x^{2}-15 x+2$$

(a) Find the remainder when $\mathrm{p}(x)$ is divided by $x+1$.

(b) (i) Show that $x+2$ is a factor of $p(x)$.$[1]$

(ii) Write $\mathrm{p}(x)$ as a product of linear factors.$[3]$


16. (CIE $0606 / 2020 / \mathrm{s} / 22 / \mathrm{q} 4)$

The three roots of $\mathrm{p}(x)=0$, where $\mathrm{p}(x)=2 x^{3}+a x^{2}+b x+c$ are $x=\frac{1}{2}, x=n$ and $x=-n$, where $a, b, c$ and $n$ are integers. The $y$-intercept of the graph of $y=p(x)$ is 4 . Find $p(x)$, simplifying your coefficients.[5]


Answer


1. (i) $a=3, b=2$

(ii) $\frac{-35}{27}$

2. (i) Show

(ii) $a=\frac{1}{2}, b=2$

3. (i) $a=10, b=43, c=36$

(ii) 21

4. (i) $a=b=6$

(ii) $(2 x-1)\left(3 x^{2}+10 x+8\right)$

(iii) $(2 x-1)(x+2)(3 x+4)$

(iv) $\theta=30^{\circ}, 150^{\circ}$

5. (i) Show (ii) $a=27, b=84$

(iii) $(2 x-1)\left(x^{2}+14 x+49\right)$

(iv) $(2 x-1)(x+7)^{2}$

6. (a) (i) Show (ii) $(2 x-1)\left(2 x^{2}+x+5\right)$

(b)(i) show (ii) Show

(ii) $x=\frac{\pi}{6}, \frac{5 \pi}{6}$

7. (i) Show

(ii) $(x-2)(2 x-3)(2 x+1), x=2,1.5,-0.5$

8. $(-0.05,-6.2)$

9. $a=3, b=2, \quad c=720$

10. (i) $p=-4$ (ii) Show

(iii) $\frac{d y}{d x}=3 x^{2}-2 x-14$

(iv) $y=40$ (v) $y=2 x+44$

11. (a) $a=-13, b=1$

(b)(i) 0 (ii) $(x-2)(2 x-1)(3 x+1)$

12. (a) $a=6, b=-24$ (

(b) $(2 x+1)\left(3 x^{2}-12\right)$

(c) $x=-\frac{1}{2}, \pm 2$

13. (a) $p(x)=(x-2)\left(6 x^{2}-11 x-10\right)$

(b) $2,-\frac{2}{3}, \frac{5}{2}$

14. (a) $A=6, B=-5, C=1$

(b) $f(x)=(x+4)(3 x-1)(2 x-1)$ (c) $-19$

15. (a) Remainder $=24$

(b)(i) Prove (ii) $p(x)=(x+2)(3 x-1)(5 x-1)$

16. $p(x)=2 x^{3}-x^{2}-8 x+4$


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