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

  Let $f(x)=x^4+5x^3+a{x}^2+bx+2$ , 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 =   

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

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

  When $p(x)=x^2-2x+p$ 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 $p(x)=2x^4+a_3{x}^3+a_2{x}^2+a_{1}x-12$ is divisible by each of (x+1),(x-1) and (x-2) .
Find the values of $a_1$,$a_2$ and $a_3$ . $a_1$ =    $a_2$    $a_3$   

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



1. Using the information shown in the diagram, find the values of a, b, c and d . a =    b =    c =    d =   
2. Let $g(x)=-\frac{3}{4}f(-2x+1)$ .
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 equatiow i5k 0f5-hfztn $32x^3-144x^2+214x-105=0$ .
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 polynomialg6 bhgix+/3ie ; y7hri $p(z)=z^5+z^4-z^3+z^2+4z+2$ , 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 $p(z)=(z+1{)}^3(z^2+cz+d)$ .
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 $p(x)=x^2-2x-c$ 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 for the polynomial etws16 u5exh 1: dj56lm7vesco quation $z^4-10z^3+c{z}^2+dz+170$ 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 equation axx nut*e9tle )f; 5;r$2z^4+a{z}^3+b{z}^2+cz+d=0$, 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}^5-2x^4-5$$x^3+40x^2+16x-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 $f(x)=2x^4+5x^3+a{x}^2+bx+4$ , 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 equati ruesag4:g37v on $x^3+b{x}^2+cx+d=0$ has three roots $x_1,x_2$ and $x_3$ . By expanding the product $(x-x_1)(x-x_2)(x-x_3)$, show that
1. 1. $b=-(x_1+x_2+x_3)$ ;
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 .