$\text { Given that }(x-4) \text { is a factor of } f(x)=x^{3}-2 x^{2}+a x+b \text { and that division of } f(x) \text { by }(x+2) \text { leaves a remainder of } 18 \text {, find the value of } a \text { and the value of } b \text {. }$a =    b =   

  Let $f(x)=x^{4}+5 x^{3}+a x^{2}+b x+2$ , for $x \in \mathbb{R}$ , where a, b are constants. The remainder when f(x) is divided by (x-1) is 6 , and the remainder when f(x) is divided by (x+2) is -6 . Find the value of a and the value of b . a =    b =   

  $\text { The quadratic equation } x^{2}-k x+(k-3)=0 \text { has roots } \alpha \text { and } \beta \text { such that } \alpha^{2}+\beta^{2}=6 \text {. Without solving the equation, find the possible values of the real number } k$      

  $\text { The quadratic equation } x^{2}-k x+(k-1)=0 \text { has roots } \alpha \text { and } \beta \text {. Without solving the equation, find the possible values of the real number } k \text { given that } \alpha^{2}+\beta^{2}=17 \text {. }$      

  When $p(x)=x^{2}-2 x+p$ is divided by x-r , the remainder is 4 .
Given that p, r $\in \mathbb{R}$ , find the largest possible value for p .   

  The polynomial $ p(x)=2 x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x-12$ is divisible by each of (x+1),(x-1) and (x-2) .
Find the values of $a_{1}$,$a_{2}$ and $a_{3}$ . $a_{1}$ =    $a_{2}$    $a_{3}$   

  $\text { Let } f(x)=a x^{3}+b x^{2}+c x+d \text {, for } x \in \mathbb{R} \text {, where } a, b, c, d \in \mathbb{Q} \text {. The diagram below shows part of the graph of } y=f(x) \text {. }$



1. Using the information shown in the diagram, find the values of a, b, c and d . a =    b =    c =    d =   
2. Let $g(x)=-\frac{3}{4} f(-2 x+1)$ .
1. Find the coordinates of the points where the graph of y=g(x) intercepts the x -axis. (x,y) x =    y =   (x,y) x =    y =  (x,y) x =    y =  
2. Find the y -intercept of the graph of y=g(x) . P (x,y) x =    y =  

  Consider the equatio mikp)2x xpjw3.i2 oi,n $32 x^{3}-144 x^{2}+214 x-105=0$ .
1. Find the sum and product of the roots of this equation. sum of roots =    product of roots =   
2. The roots of this equation are three consecutive terms of an arithmetic sequence. Taking the roots to be a-d, a, a+d , solve the equation.   

  Consider the polynomiarf3;,wub,66wr ek dab okf.e /l $p(z)=z^{5}+z^{4}-z^{3}+z^{2}+4 z+2$ , for $z \in \mathbb{C}$ .
1. Write down the sum and product of the roots of p(z)=0 .sum of roots =    product of roots =   
2. Show that (z+1) is a factor of p(z) .

The polynomial can be written in the form $p(z)=(z+1)^{3}\left(z^{2}+c z+d\right)$ .
3. Find the value of c and the value of d .d =    c =   
4. Hence find the complex roots of p(z)=0 .    $\pm \mathrm{i}$

  When $ p(x)=x^{2}-2 x-c$ is divided by (x-r) , the remainder is -5 .
Given that c, r $\in \mathbb{R}$ , find the smallest possible value for c .   

  Two distinct roots fodq e260pzckmped9t4 -r the polynomial equation $z^{4}-10 z^{3}+c z^{2}+d z+170 $ are $a+\mathrm{i}$ and $1+\mathrm{i}$ b where $a, b, c, d \in \mathbb{Z}$, b>0 .
1. Write down the other two roots in terms of a and b .
2. Find the value of a and the value of b . a =    b =   

Consider the equatiopn 3uwr1n) u,kn $2 z^{4}+a z^{3}+b z^{2}+c z+d=0 $, where $a, b, c, d \in \mathbb{R}$ and $z \in \mathbb{C}$ . Two of the roots of the equation are $\log _{2}$ 10 and $\mathrm{i} \sqrt{5}$ and the sum of all the roots is $4+\log _{2} 5$ .
Show that 15 a+d+90=0 .

  Let $p(x)=\frac{1}{4} x^{5}-2 x^{4}-5$$ x^{3}+40 x^{2}+16 x-128 $, for x $\in \mathbb{R}$ .
1. For the polynomial equation p(x)=0 , state:
1. the sum of the roots;   
2. the product of the roots.   

A new polynomial is defined by q(x)=p(2 x-2) .
2. For the polynomial equation q(x)=0 , find:
1. the sum of the roots;   
2. the product of the roots.   

It is given that $f(x)=2 x^{4}+5 x^{3}+a x^{2}+b x+4$ , for $x \in \mathbb{R}$ , where $a, b \in \mathbb{Z}^{+}$ .
1. Given that $x^{2}+x-2$ is a factor of f(x) , find the values of a and b .
2. Factorise f(x) into a product of linear factors.
3. Sketch the graph of y=f(x) , labeling the maximum and minimum points and the x and y intercepts.
4. Using your graph, state the range of values of c for which f(x)=c has exactly four distinct real roots.

The cubic polynomial equatiop7oca8y7 ) d2 qfmsqh1n $ x^{3}+b x^{2}+c x+d=0$ has three roots $x_{1}, x_{2} $ and $x_{3}$ . By expanding the product $\left(x-x_{1}\right)\left(x-x_{2}\right)\left(x-x_{3}\right) $, show that
1. 1. $ b=-\left(x_{1}+x_{2}+x_{3}\right)$ ;
2. $ c=x_{1} x_{2}+x_{1} x_{3}+x_{2} x_{3} $
3. $d=-x_{1} x_{2} x_{3}$ .

It is given that b=-9 and c=45 for parts (b) and (c) below.
2. 1. In the case that the three roots $x_{1}, x_{2}$ and $ x_{3} $ form an arithmetic sequence, show that one of the roots is 3 .
2. Hence determine the value of d .
3. In another case the three roots form a geometric sequence. Determine the value of d .