(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(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) \((p\,q)(x)=2x^6+x^5+7x^4+11x^3+13x^2+11x-35\)(a) at \(x=1\)
(b) at \(x=2\)
(c) at \(x=-4\)
Solutions: (a) \(p(1)=-3\); (b) \(p(2)=9\); (c) \(p(-4)=27\)(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 \((p(4)=0)\)(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) \(x_1=0,~ x_2=1,~ x_3=-1,~ x_4=3\)(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) \(x_{1,2,3}=0,~ x_{4,5}=2\)(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) \(p(x)=x^2(x-5)(x^2+4)\)(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) \(x_1=0,~ x_2=1,~ x_3=-1,~ x_4=\sqrt{3},~ x_5=-\sqrt{3}\)(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) \(x_{1,2,3}=0,~ x_{4,5}=-2\)(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) \(p(x)=(x-\sqrt{3})(x+\sqrt{3})(x^2+1)\)(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 \(P_2(1,4)\)(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 \(P_2(1,-1)\)(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) \(\cdots=-\frac{3}{2}\)(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) \(\cdots=3\)(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.(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: \(y=\frac{1}{2}\);(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: \(y=0\);(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 \(=\mathbb{R}\setminus\{\frac{3}{2}\}\)(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 \(=\mathbb{R}\setminus\{-1\}\)(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: \(P(2,1)\)(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: \((-0.5,-3)\)(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: \((-2,-0.25)\)(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: \((4.162,0.075)\)(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: \((-1.387,2.081)\)(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(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: \(x_1=1-\sqrt{2},~ x_2=1+\sqrt{2}\)