Domov

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)   \((p\,q)(x)\)

    Solutions:    (a)  \((p+q)(x)=3x^3+x^2+5x+2\);     (b)  \((p-q)(x)=x^3+x^2+x-12\);     (c)  \((p\,q)(x)=2x^6+x^5+7x^4+11x^3+13x^2+11x-35\)
  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: \(x_1\approx1.755\);     (b)  maximum \(P_1(0.333,-0.852)\), minimum \(P_2(1,-1)\)

Limits

  1. Evaluate the following limits:

    (a)   \({\displaystyle\lim_{x\to\infty}\frac{2x+3}{x-1}}\)

    (b)   \({\displaystyle\lim_{x\to\infty}\frac{x+2}{3x+5}}\)

    (c)   \({\displaystyle\lim_{x\to-\infty}\frac{5+6x}{1-4x}}\)

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

    (a)   \({\displaystyle\lim_{x\to\infty}\frac{x^2+x+1}{x^2+5x}}\)

    (b)   \({\displaystyle\lim_{x\to-\infty}\frac{2x^2-x}{x^2+1}}\)

    (c)   \({\displaystyle\lim_{x\to\pm\infty}\frac{(x+1)^2}{2x(x+2)}}\)

    (d)   \({\displaystyle\lim_{x\to\pm\infty}\frac{3x^3+1}{(x+1)^3}}\)

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

    (a)   \({\displaystyle\lim_{x\to\infty}\frac{x^2+2x}{x^3+1}}\)

    (b)   \({\displaystyle\lim_{x\to\infty}\frac{1}{x^2+1}}\)

    (c)   \({\displaystyle\lim_{x\to\infty}\frac{x^2}{2x+3}}\)

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

Rational functions

  1. 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: \(y=\frac{1}{2}\);
  2. 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: \(y=0\);
  3. 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 \(=\mathbb{R}\setminus\{\frac{3}{2}\}\)
  4. 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 \(=\mathbb{R}\setminus\{-1\}\)
  5. 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: \(P(2,1)\)
  6. 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: \((-0.5,-3)\)
  7. 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: \((-2,-0.25)\)
  8. 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: \((4.162,0.075)\)
  9. 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: \((-1.387,2.081)\)
  10. 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
  11. 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 \(x_1\approx-0.414,~ x_2\approx2.414\), vertical asymptotes \(x=1\) and \(x=-1\), horizontal asymptote \(y=1\);     (c)  zeros: \(x_1=1-\sqrt{2},~ x_2=1+\sqrt{2}\)

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