Ch. 3 Review Exercises - Precalculus | OpenStax (2024)

1.

$\left(4+3i\right)+\left(-2-5i\right)$

2.

$\left(6-5i\right)-\left(10+3i\right)$

3.

$\left(2-3i\right)\left(3+6i\right)$

4.

$\frac{2-i}{2+i}$

5.

$x^2-4x+5=0$

6.

$x^2+2x+10=0$

7.

$f(x)=x^2-4x-5$

8.

$f(x)=-2x^2-4x$

9.

The vertex is $(–2,3)$ and a point on the graph is $(3,6).$

10.

The vertex is $(–3,6.5)$ and a point on the graph is $(2,6).$

11.

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.

12.

An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height, $h,$ in terms of horizontal distance traveled, $x,$ given by $h(x)=\frac{-32}{{(120)}^2}{x}^2+x.$ Find the maximum height the object attains.

13.

$f(x)=4x^5-3x^3+2x-1$

14.

$f(x)=5^{x+1}-x^2$

15.

$f(x)=x^2\left(3-6x+x^2\right)$

16.

$f(x)=2x^4+3x^3-5x^2+7$

17.

$f(x)=4x^3-6x^2+2$

18.

$f(x)=2x^2(1+3x-x^2)$

19.

$f(x)={(x+3)}^2(2x-1){(x+1)}^3$

20.

$f(x)=x^5+4x^4+4x^3$

21.

$f(x)=x^3-4x^2+x-4$

22.

23.

24.

Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the function $f(x)=x^3-5x+1$

25.

$\frac{x^3-2x^2+4x+4}{x-2}$

26.

$\frac{3x^4-4x^2+4x+8}{x+1}$

27.

$\frac{x^3-2x^2+5x-1}{x+3}$

28.

$\frac{x^3+4x+10}{x-3}$

29.

$\frac{2x^3+6x^2-11x-12}{x+4}$

30.

$\frac{3x^4+3x^3+2x+2}{x+1}$

31.

$2x^3-3x^2-18x-8=0$

32.

$3x^3+11x^2+8x-4=0$

33.

$2x^4-17x^3+46x^2-43x+12=0$

34.

$4x^4+8x^3+19x^2+32x+12=0$

35.

$x^3-3x^2-2x+4=0$

36.

$2x^4-x^3+4x^2-5x+1=0$

37.

$f(x)=\frac{x+2}{x-5}$

38.

$f(x)=\frac{x^2+1}{x^2-4}$

39.

$f(x)=\frac{3x^2-27}{x^2+x-2}$

40.

$f(x)=\frac{x+2}{x^2-9}$

41.

$f(x)=\frac{x^2-1}{x+2}$

42.

$f(x)=\frac{2x^3-x^2+4}{x^2+1}$

43.

$f(x)={(x-2)}^2,x\ge 2$

44.

$f(x)={(x+4)}^2-3,x\ge -4$

45.

$f(x)=x^2+6x-2,x\ge -3$

46.

$f(x)=2x^3-3$

47.

$f(x)=\sqrt{4x+5}-3$

48.

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

49.

$y$varies directly as the square of $x.$If when $x=3,y=36,$ find $y$ if $x=4.$

50.

$y$varies inversely as the square root of $x$If when $x=25,y=2,$ find $y$ if $x=4.$

51.

$y$ varies jointly as the cube of $x$ and as $z.$ If when $x=1$ and $z=2,$ $y=6,$ find $y$ if $x=2$ and $z=3.$

52.

$y$ varies jointly as $x$ and the square of $z$ and inversely as the cube of $w.$ If when $x=3,$ $z=4,$ and $w=2,$ $y=48,$ find $y$ if $x=4,$ $z=5,$ and $w=3.$

53.

The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

54.

The volume $V$ of an ideal gas varies directly with the temperature $T$ and inversely with the pressure $P$. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.