Polynomials and rational functions

Polynomials

1. Write down the degree, the leading coefficient and the constant term of the following polynomials:
   

   (a)   $p(x)=4x^3-5x^2+3x+12$

   (b)   $p(x)=\frac{1}{2}{x}^5-x^2-3$

   (c)   $p(x)=-x^6+x^4-x^2$

   Solutions:    (a)  degree = 3, leading coefficient = 4, constant term = 12;     (b)  degree = 5, leading coefficient $=\frac{1}{2}$, constant term $=-3$;     (c)  degree = 6, leading coefficient $=-1$, constant term = 0
2. Given polynomials $p(x)=2x^3+x^2+3x-5$ and $q(x)=x^3+2x+7$ write down:
   

   (a)   $(p+q)(x)$

   (b)   $(p-q)(x)$

   (c)   $(pq)(x)$

   Solutions:    (a)  $(p+q)(x)=3x^3+x^2+5x+2$;     (b)  $(p-q)(x)=x^3+x^2+x-12$;     (c)  $(pq)(x)=2x^6+x^5+7x^4+11x^3+13x^2+11x-35$
3. Find the value of the polynomial $p(x)=x^4+2x^3-6x^2+x-1$ at given points:
   

   (a)   at $x=1$

   (b)   at $x=2$

   (c)   at $x=-4$

   Solutions:    (a)  $p(1)=-3$;     (b)  $p(2)=9$;     (c)  $p(-4)=27$
4. Which of the following numbers is a zero of the polynomial $p(x)=x^3-3x^2-10x+24$:
   

   (a)   $x=1$

   (b)   $x=2$

   (c)   $x=3$

   (d)   $x=4$

   Solutions:    (a)  $x=1$ is not a zero $(p(1)=12)$;     (b)  $x=2$ is a zero $(p(2)=0)$;     (c)  $x=3$ is not a zero $(p(3)=-6)$;     (d)  $x=4$ is a zero $(p(4)=0)$
5. Find zeros of the following polynomials using factorisation:
   

   (a)   $p(x)=x^3+2x^2-15x$

   (b)   $p(x)=x^3-5x^2-4x+20$

   (c)   $p(x)=x^4-3x^3-x^2+3x$

   Solutions:    (a)  $x_1=0,x_2=3,x_3=-5$;     (b)  $x_1=2,x_2=-2,x_3=5$;     (c)  $x_1=0,x_2=1,x_3=-1,x_4=3$
6. Find zeros of the following polynomials using factorisation:
   

   (a)   $p(x)=x^3-2x^2-4x+8$

   (b)   $p(x)=x^4-4x^2$

   (c)   $p(x)=x^5-4x^4+4x^3$

   Solutions:    (a)  $x_{1,2}=2,x_3=-2$;     (b)  $x_{1,2}=0,x_3=2,x_4=-2$;     (c)  $x_{1,2,3}=0,x_{4,5}=2$
7. Factorise the following polynomials:
   

   (a)   $p(x)=x^5-3x^4-5x^3+15x^2+4x-12$

   (b)   $p(x)=x^5-2x^4-3x^3+6x^2-4x+8$

   (c)   $p(x)=x^5-5x^4+4x^3-20x^2$

   Solutions:    (a)  $p(x)=(x+1)(x-1)(x+2)(x-2)(x-3)$;     (b)  $p(x)=(x-2{)}^2(x+2)(x^2+1)$;     (c)  $p(x)=x^2(x-5)(x^2+4)$
8. Find zeros and draw graphs of the following polynomials:
   

   (a)   $p(x)=x^3+x^2-2x$

   (b)   $p(x)=x^3-3x$

   (c)   $p(x)=x^4-5x^2+4$

   (d)   $p(x)=x^5-4x^3+3x$

   Solutions:    (a)  $x_1=0,x_2=1,x_3=-2$;     (b)  $x_1=0,x_2=\sqrt{3},x_3=-\sqrt{3}$;     (c)  $x_1=1,x_2=-1,x_3=2,x_4=-2$;     (d)  $x_1=0,x_2=1,x_3=-1,x_4=\sqrt{3},x_5=-\sqrt{3}$
9. Find zeros and draw graphs of the following polynomials:
   

   (a)   $p(x)=x^4-5x^3+6x^2$

   (b)   $p(x)=x^4+x^3-2x^2$

   (c)   $p(x)=x^4-2x^3$

   (d)   $p(x)=x^5+4x^4+4x^3$

   Solutions:    (a)  $x_{1,2}=0,x_3=2,x_4=3$;     (b)  $x_{1,2}=0,x_3=1,x_4=-2$;     (c)  $x_{1,2,3}=0,x_4=2$;     (d)  $x_{1,2,3}=0,x_{4,5}=-2$
10. Factorise and draw graphs of the following polynomials:
    

    (a)   $p(x)=x^3+3x^2+x+3$

    (b)   $p(x)=x^4-2x^2-3$

    Solutions:    (a)  $p(x)=(x+3)(x^2+1)$;     (b)  $p(x)=(x-\sqrt{3})(x+\sqrt{3})(x^2+1)$
11. Use your GDC to draw graph of the polynomial $p(x)=2x^3-3x^2+5$
    

    (a)   Using GDC find zeros.

    (b)   Using GDC find extreme points (maxima and minima).

    Solutions:    (a)  zero: $x_1=-1$;     (b)  maximum $P_1(0,5)$, minimum $P_2(1,4)$
12. Use your GDC to draw graph of the polynomial $p(x)=x^3-2x^2+x-1$
    

    (a)   Using GDC find zeros.

    (b)   Using GDC find extreme points (maxima and minima).

    Solutions:    (a)  zero: $x_1\approx 1.755$;     (b)  maximum $P_1(0.333,-0.852)$, minimum $P_2(1,-1)$

Limits

13. Evaluate the following limits:
    

    (a)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{2x+3}{x-1}}$

    (b)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x+2}{3x+5}}$

    (c)   ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{5+6x}{1-4x}}$

    Solutions:    (a)  $\cdots =2$;     (b)  $\cdots =\frac{1}{3}$;     (c)  $\cdots =-\frac{3}{2}$
14. Evaluate the following limits:
    

    (a)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x^2+x+1}{x^2+5x}}$

    (b)   ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{2x^2-x}{x^2+1}}$

    (c)   ${\displaystyle \underset{x\to \pm \mathrm{\infty }}{lim}\frac{(x+1{)}^2}{2x(x+2)}}$

    (d)   ${\displaystyle \underset{x\to \pm \mathrm{\infty }}{lim}\frac{3x^3+1}{(x+1{)}^3}}$

    Solutions:    (a)  $\cdots =1$;     (b)  $\cdots =2$;     (c)  $\cdots =\frac{1}{2}$;     (d)  $\cdots =3$
15. Evaluate the following limits (if possible):
    

    (a)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x^2+2x}{x^3+1}}$

    (b)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{1}{x^2+1}}$

    (c)   ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x^2}{2x+3}}$

    Solutions:    (a)  $\cdots =0$;     (b)  $\cdots =0$;     (c)  Not possible – the limit doesn't exist.

Rational functions

16. Write down zeros, vertical asymptotes and horizontal asymptotes of the following functions and draw the graphs:
    

    (a)   ${\displaystyle f(x)=\frac{x+1}{x-1}}$

    (b)   ${\displaystyle f(x)=\frac{2x-3}{x-1}}$

    (c)   ${\displaystyle f(x)=\frac{x+1}{2x+5}}$

    Solutions:    (a)  zero: $x=-1$, vertical asymptote (pole): $x=1$, horizontal asymptote: $y=1$;     (b)  zero: $x=\frac{3}{2}$, vertical asymptote (pole): $x=1$, horizontal asymptote: $y=2$;     (c)  zero: $x=-1$, vertical asymptote (pole): $x=-\frac{5}{2}$, horizontal asymptote: $y=\frac{1}{2}$;
17. Write down zeros, vertical asymptotes and horizontal asymptotes of the following functions and draw the graphs:
    

    (a)   ${\displaystyle f(x)=\frac{1}{x+2}}$

    (b)   ${\displaystyle f(x)=\frac{2}{3x-4}}$

    (c)   ${\displaystyle f(x)=\frac{1}{1-x}}$

    Solutions:    (a)  zero: /, vertical asymptote (pole): $x=-2$, horizontal asymptote: $y=0$;     (b)  zero: /, vertical asymptote (pole): $x=\frac{4}{3}$, horizontal asymptote: $y=0$;     (c)  zero: /, vertical asymptote (pole): $x=1$, horizontal asymptote: $y=0$;
18. Given the function ${\displaystyle f(x)=\frac{3x-3}{2x-1}}$
    

    (a)   write down the zero, vertical asymptote and horizontal asymptote,

    (b)   draw the graph,

    (c)   write the domain and range.

    Solutions:    (a)  zero: $x=1$, vertical asymptote: $x=\frac{1}{2}$, horizontal asymptote: $y=\frac{3}{2}$;     (c)  domain $=\mathbb{R}\setminus \{\frac{1}{2}\}$, range $=\mathbb{R}\setminus \{\frac{3}{2}\}$
19. Given the function ${\displaystyle f(x)=\frac{1-x}{2+x}}$
    

    (a)   write down the zero, vertical asymptote and horizontal asymptote,

    (b)   draw the graph,

    (c)   write the domain and range.

    Solutions:    (a)  zero: $x=1$, vertical asymptote: $x=-2$, horizontal asymptote: $y=-1$;     (c)  domain $=\mathbb{R}\setminus \{-2\}$, range $=\mathbb{R}\setminus \{-1\}$
20. A rational function has the equation ${\displaystyle f(x)=\frac{x^2}{x^2+x-2}}$.
    

    (a)   Find zeros, vertical asymptotes and horizontal asymptote.

    (b)   Draw the graph.

    (c)   Find the point where the graph intersects the horizontal asymptote.

    Solutions:    (a)  zero $x_{1,2}=0$, vertical asymptotes $x=1$ and $x=-2$, horizontal asymptote $y=1$;     (c)  intersection: $P(2,1)$
21. A rational function has the equation ${\displaystyle f(x)=\frac{x^2+4x+4}{x^2-1}}$.
    

    (a)   Find zeros, vertical asymptotes and horizontal asymptote.

    (b)   Draw the graph.

    (c)   Find the point where the graph intersects the horizontal asymptote.

    (d)   Find the extreme points.

    Solutions:    (a)  zero $x_{1,2}=-2$, vertical asymptotes $x=1$ and $x=-1$, horizontal asymptote $y=1$;     (c)  intersection: $P(-1.2,1)$;     (d)  min: $(-2,0)$, max: $(-0.5,-3)$
22. A rational function has the equation ${\displaystyle f(x)=\frac{x+1}{x^2}}$.
    

    (a)   Find zeros, vertical asymptotes and horizontal asymptote.

    (b)   Draw the graph.

    (c)   Find the extreme points.

    Solutions:    (a)  zero $x_1=-1$, vertical asymptotes $x_{1,2}=0$, horizontal asymptote $y=0$;     (c)  min: $(-2,-0.25)$
23. A rational function has the equation ${\displaystyle f(x)=\frac{x-1}{x^2+5x+4}}$.
    

    (a)   Find zeros, vertical asymptotes and horizontal asymptote.

    (b)   Draw the graph.

    (c)   Find the extreme points.

    Solutions:    (a)  zero $x_1=1$, vertical asymptotes $x=-1$ and $x=-4$, horizontal asymptote $y=0$;     (c)  min: $(-2.162,1.481)$, max: $(4.162,0.075)$
24. A rational function has the equation ${\displaystyle f(x)=\frac{x^2-3x}{x^2+1}}$.
    

    (a)   Find zeros, vertical asymptotes and horizontal asymptote.

    (b)   Draw the graph.

    (c)   Find the extreme points.

    Solutions:    (a)  zeros $x_1=0,x_2=3$, vertical asymptotes don't exist, horizontal asymptote $y=1$;     (c)  min: $(0.721,-1.081)$, max: $(-1.387,2.081)$
25. A rational function has the equation ${\displaystyle f(x)=\frac{x^3-2x^2}{x^2-2x+1}}$.
    

    (a)   Find zeros, vertical asymptotes and horizontal asymptote.

    (b)   Draw the graph.

    Solutions:    (a)  zeros $x_{1,2}=0,x_3=2$, vertical asymptote $x=1$, horizontal asymptote doesn't exist
26. A rational function has the equation ${\displaystyle f(x)=\frac{x}{x+1}-\frac{1}{x-1}}$.
    

    (a)   Draw the graph (using GDC).

    (b)   Write down zeros, vertical asymptotes and horizontal asymptote (using GDC).

    (c)   Find the zeros algebraically and write down the exact values.

    Solutions:    (b)  zeros $x_1\approx -0.414,x_2\approx 2.414$, vertical asymptotes $x=1$ and $x=-1$, horizontal asymptote $y=1$;     (c)  zeros: $x_1=1-\sqrt{2},x_2=1+\sqrt{2}$

 Index