Sample Test on Review Material (Chapters P,1 and 2)

1. Which of the following numbers are rational? a) $\frac{-4}{7}\qquad$b) $\sqrt{9}\qquad$c) $.10110111011110111110...%% \qquad$d) $\sqrt{17}\qquad$e) $\frac{\pi }{4}$ f) $(\sqrt[3]{-5})^{6}\qquad$g) $.123456789\qquad$h) $2.123333...$

2. a) Express $x$ as a rational number: $x=1.035353535...$ b) Express $x$ as a repeating decimal: $x=\frac{2}{7}$

3. Name the basic property of the real numbers which justifies each of the following: a) $5x-3x=2x\qquad$b) $(3x)y=3(xy)\qquad$c) $3x+0=3x$

4. Evaluate each of the following: a) $48\div 6-2\times \left( 5-1\right) \qquad$b) $%% (6+2^{3}\times 5\div 4)\div 2^{2}+4\qquad$c) $\left\vert 3-2\left( 9-5\right) \right\vert$ d) $\frac{48}{-12}\qquad$e) $\frac{-2(5-7)}{3(6-12)}\qquad$ f) $\frac{3^{-2}}{9^{\frac{-1}{2}}}\qquad$g) $\frac{\sqrt{25\cdot 64}} {\sqrt{16\cdot 100}}$

5. Find a value of $x$ so that $\left\vert 2+x\right\vert \neq \left\vert 2\right\vert +\left\vert x\right\vert$

6. In each of the following, find all values of $x$ for which the statement is true. a) $\left\vert x-3\right\vert =9\qquad$b) $\frac{4}{12}=\frac{9}{x}$

7. Simplify each of the following: a) $\frac{-9xy}{6y}\qquad$b) $\frac{4x-12}{9-3x}\qquad$c) $\frac{x-5} {3}+\frac{4x-3}{5}-x\qquad$d) $\frac{3}{2x-7}-\frac{5}{x+1}\qquad$ e) $\frac{2x+3}{3x-9}\cdot \frac{x-3}{4x+6}\qquad$ f) $\frac{\frac{5x-7}{4x+2}}{\frac{6x}{2x+1}}\qquad$ g) $\frac{\frac{4}{x-2}-\frac{3}{x-3}}{\frac{1}{x-3}+\frac{2}{x-2}}\qquad$ h) $\frac{(a^{2}b)((4ab^{-1})^{3}}{(2ab)^{5}}\qquad$ i) $\left( \frac{x^{-1}y^{2}}{xy^{3}}\right) ^{-2}$

8. Simplify: a) $(5x-4)+(2x^{2}-2x+1)-(x^{2}-3)$

9. Perform each of the following multiplications: a) $(x^{2}+5y^{2})(2x-3y)$         b) $(a+2b)(a+2b)$          c) $2x(x-3)((5x+1)(x^{2}-2x+5)$

10. Perform each of the following divisions to get a quotient and remainder. a) $(x^{2}+3x-1)\overline{)x^{5}-6x^{4}+2x^{2}+12x-7}\qquad$ b) $(x-5)\overline{)x^{4}-9x^{3}-16x-10}$

Sample Test on Review Material (Chapters P,1 and 2) (continued)

11. Write $\frac{x^{5}-6x^{4}+2x^{2}+12x-7}{x^{2}+3x-1}$ as the sum of a polynomial and a rational function (i.e., a fractional expression) in which the degree of the numerqator is less than the degree of the denominator.

12. i) Let $P(x)=\allowbreak 5x^{3}-8x^{2}-7x+6$. Find each of the following: a) $P(0)\qquad$b) $P(1)$        c) $P(-1)\qquad$ d) $P(2)\qquad$e) $P(-2)\qquad$ f) $P(-1+\sqrt{2})$ ii) Find all the roots of $P(x)$, and explain how you know that you have them all. See section 10.1 of the text.

13. Factor each of the following: a) $4x^{2}-9\qquad \qquad$b) $16x^{2}+8x+1\qquad \qquad$c) $25x^{2}-20xy+4y^{2}\qquad \qquad$ d) $4y^{2}-12y+5xy-15x \qquad \qquad \qquad$e) $10x^{2}+x-3$ f) $6t^{2}+7t-20\qquad\qquad$g) $8x^{3}-27\qquad\qquad$h) $x^{3}-x^{2}-5x-3$

14. Simplify each of the following: a) $\frac{2x+5}{x^{2}-7x+12}-\frac{x-1}{x^{2}-3x}+5\qquad$b) $\frac{5x}{x^{2}+1}-\frac{3}{x+1}\qquad$c) $\frac{x^{2}+4x}{x^{2}+4x+4}\div \frac{x^{2}-x-20}{x^{2}+2x}$ d) $\frac{\frac{x-1}{x^{2}-3x-10}-\frac{x-2}{x^{2}+2x}}{\frac{x+1}{x^{2}-5x}%% +\frac{1}{x^{2}+4x+4}}$

15. Simplify each of the following: a) $\left( \frac{-64}{125}\right) ^{-\frac{2}{3}}\qquad$b) $\left( \frac{(x^{3}y^{-1})(x^{-\frac{3}{2}}y^{\frac{1}{2}})}%% {(xy^{-1})^{3}}\right)^{2}\qquad$

16. Simplify by factioring: $(xy^{5})^{\frac{3}{5}}+x^{\frac{8}{5}}$

17. Simplify each of the following: (In each case, assume that $x$ is such that all radicals are defined.) a) $3\sqrt{x-2}+x^{2}\sqrt{x-2}-(4x-1)\sqrt{x-2}$ b) $\sqrt{x^{2}+3}-\frac{x^{2}}{\sqrt{x^{2}+3}}$

18. In each of the following, rationalize the denominator: a) $\frac{2}{\sqrt{7}}\qquad$b) $\frac{2x+1}{\sqrt{x-1}}\qquad$c) $\frac{2}{3+\sqrt{5}}\qquad$d) $\frac{x}{\sqrt[3]{(x-1)^{2}}}$

19. Find each of the following binomial coefficients: a) $_{8}C_{5}\qquad$b ) $_{37}C_{37}\qquad$c) $_{37}C_{36}$

Sample Test on Review Material (Chapters P,1 and 2) (continued)

20. Find the term in the expansion of $(2x-y)^{8}$ for which the exponent of $x$ is a) $1\qquad$b) $3\qquad$c) $5\qquad$d) $8$

21. Expand $(x-3)^{5}$ completely.

22. Solve each of the following: a) $\frac{3}{4}x-5(x- \frac{2}{3})=2x+5 \qquad$ b) $6(x-3)+3(2x-1)=9(x-7)+2(3x+6)$ c) $\frac{2}{5x-1}=\frac{3}{4+7x}\qquad\qquad$ d) $\frac{3x+1}{5x-3}=\frac{6x-7}{10x+9}\qquad\qquad$ e) $\frac{5}{x+1}-\frac{2}{x+3}=\frac{3x+8}{x^{2}+4x+3}$ f) $\frac{5}{x+1}-\frac{2}{x+3}=\frac{x-7}{x^{2}+4x+3}\qquad$ g) $\frac{5}{x+2}-\frac{2}{x-2}=\frac{x-10}{x^{2}-4}\qquad$ h) $\sqrt{5x-4}+14=0$ i) $\sqrt{5x-4}-14=0\qquad\qquad\qquad$j) $\sqrt{4x-1}+3=4x+10$ k) $\sqrt{4x-1}+5=\sqrt{4x+10}\qquad \qquad$l) $\left\vert x+5\right\vert -8=0$ m) $\left\vert x+5\right\vert +8=0\qquad \qquad \qquad \qquad$n) $\left\vert \frac{4x-3}{x+5}\right\vert =3$

23. Solve each of the following: a) $2x+7<5x-8\qquad \qquad$b) $3(x+4)\geq 5(x-7)$ c) $\frac{x+1}{2x-1}\geq 0\qquad \qquad \qquad$d) $\frac{3x+10}{x+2} \leq 2\qquad \qquad$e) $\frac{3x+5}{x+2}\leq 2\qquad \qquad$ f) $\left\vert 4x+8\right\vert <15\qquad \qquad$g) $\ \left\vert 5x+6\right\vert \leq 10\qquad \qquad$h) $\left\vert 3x-9\right\vert \geq 6$

24. Let $P_{1}(0,0)$, $P_{2}(3,4)$, and $P_{3}(7,1)$ be three points in the plane. a) Find each of the following distances         i) $d(P_{1},P_{2})$ ii) $d(P_{2},P_{3})$ iii) $d(P_{1},P_{2})$ b) Use distances to show that triangle $P_{1}P_{2}P_{3}$ is a right triangle. c) Find a point $P_{4}$ so that $P_{1}P_{2}P_{3}P_{4}$ is a square.

25. Using the same three points as in problem 24, a) Find each of the following midpoints:         i) $M_{12}$, the midpoint of $P_{1}$ and $P_{2}$;         ii) $M_{13}$, the midpoint of $P_{1}$ and $P_{3}$;         iii) $M_{23}$, the midpoint of $P_{2}$ and $P_{3}$;
b) Find the distance $d(M_{12},M_{23})$ and show that it is half of $d(P_{1},P_{3})$ c) Show that triangle $M_{1}M_{2}M_{3}$ is a right triangle.

Sample Test on Review Material (Chapters P,1 and 2) (continued)

26. Let $P_{1}$, $P_{2}$, $P_{3}$, $M_{12}$, $M_{13}$ and $M_{23}$ be as in problems 24 and 25. a) Find each of the following slopes:         i) The slope of the line through $P_{1}$ and $P_{2}$;         ii) The slope of the line through $P_{1}$ and $P_{2}$;         iii) The slope of the line through $P_{2}$ and $P_{3}$;         iv) The slope of the line through $M_{12}$ and $M_{23}$; b) Show that the line $\overline{M_{12}M_{23}}$ is parallel to the line $\overline{P_{1}P_{3}}$. c) Use slopes to show that triangle $P_{1}P_{2}P_{3}$ is a right triangle. d) Find a point $P_{4}$ so that $P_{1}P_{3}P_{2}P_{4}$ is a parallelogram.

27. Find the equation (in any form), the slope (if one exists) and the y-intercept (if any) of each of the following lines: a) The line through the points $P(3,7)$ and $Q(5,16).$ b) The line through the points $P(3,7)$ and $Q(5,3).$ c) The line through the points $P(3,7)$ and $Q(5,7).$ d) The line through the points $P(3,7)$ and $Q(3,9).$ e) The line through $P(2,-1)$ which has slope $5.$ f) The line through $P(2,-1)$ which has slope $-2.$ g) The horizontal line through $P(2,-1).$ h) The vertical line through $P(2,-1).$ i) The line through $P(2,-1)$ which has $y$-intercept $8.$ j) The line through $P(2,-1)$ which has $x$-intercept $8.$
k) The line through $P(2,-1)$ which is parallel to the line $%% 4x+2y=11$. l) The line parallel to the line $4x+2y=11$ which had $y$-intercept $8$. m) The line perpendicular to the line $4x+2y=11$ which had $y$-intercept $8$. n) The perpendicular bisector of the line through $P(3,7)$ and $Q(5,7).$

28. Find the point(s) of intersection (if they exist) of the following pairs of lines: a) $x+y=7$ and $4x+5y=33$         b) $2x-5y=4$ and $%% 3x+2y=6$ c) $3x-4y=18$ and $8y-6x=4\qquad$d) $3x-4y=18$ and $%% y=-5x+7$ e) $3x-6y=21$ and $y=\frac{1}{2}x-\frac{7}{2}$

Sample Test on Review Material (Chapters P,1 and 2) (continued)

29. Solve the following: a) Sally bought a blouse a skirt and a sweater at a sale. If the blouse was marked one third off and Sally paid $\$12$, the skirt was marked $25\%$ off and Sally paid $\$21$, and the sweater was $50\%$ off and Sally paid $\$18$, to the nearest tenth of a percent, what was the overall percentage discount that Sally received for buying the three items? b) Mary figures that if she can average $45$ miles per hour for her trip, then she can arrive at her destination just in time for the ceremony. So far, she has driven $120$ miles and averaged $40$ miles per hour. If she still has $60$ miles to go, what speed must she average for the remainder of her trip in order to arrive at the time she had planned?