Math  2  –  Chapter  2  –  Test  Review  Sheet    Students  Will  Be  Able  To:    

• Use  basic  rules  of  algebra  to  transform  expressions.  (TB  2.02,  2.04)  • Determine  whether  expressions  are  equivalent.  (TB  2.02)  • Factor  expressions  by  finding  a  common  factor.  (TB  2.03)  • Apply  the  zero-­‐product  property  to  factored  expressions.  (TB  2.03,  2.04)  • Recognize  and  provide  examples  of  polynomials.  (TB  2.06)  • Use  the  terms  coefficient,  constant,  linear,  quadratic,  cubic,  monomial,  binomial,  

trinomial,  and  degree  to  solve  problems  (TB  2.06)  • Expand  polynomials  and  express  them  in  standard  form  (normal  form)  (TB  2.07)  • Determine  whether  polynomials  in  different  forms  are  equivalent  (TB  2.07)  • Add,  subtract,  and  multiply  polynomials  (TB  2.08)  • Factor  the  difference  of  squares  (TB  2.10)  • Factor  any  monic  polynomial  that  is  factorable  over  ℤ  (the  integers)  (TB  2.11)  • Factor  any  polynomial  that  is  factorable  over  ℝ  (the  real  numbers)  by  completing  

the  square.  (TB  2.12)  • Combine  factoring  tools  to  completely  factor  more  complex  polynomials  (TB  2.12)  

 Practice  Problems:    

1. Factor  each  of  the  following.    

 2. Which  identity  did  you  use  in  question  1?  State  the  identity.  Then  pick  one  of  the  

expressions  and  show  how  it  follows  the  identity.    

3. Factor  each  quadratic  expression  over  ℤ.  If  you  cannot  factor  an  expression,  explain  why.  

   

4. Factor  each  quadratic  expression  over  ℤ  by  completing  the  square.  

   

5. Solve  each  equation.  

   

6. Prove  that  the  following  is  an  identity.  𝑥 + 3 𝑥! − 5𝑥 + 4 = 𝑥! − 3𝑥! − 11𝑥 + 12    

7. Is  the  following  an  identity?  Support  your  answer  by  showing  work.  2𝑚!𝑛 −𝑚! −𝑚𝑛! − 2𝑚𝑛 +𝑚! + 𝑛! = 𝑚 − 1 𝑚 − 𝑛 𝑛 −𝑚  

 

 

8. A  square  has  side  length  (k  –  x).  A  smaller  square  with  side  length  x  is  cut  from  a  corner.  

a. Make  a  diagram  to  represent  this  situation.  b. Find  the  area  of  the  leftover  shape  in  terms  of  k  and  x.  

 9. Find  all  the  solutions  to  each  equation.  Use  factoring  and  ZPP.  

a.   𝑥 − 7 2𝑥 + 1 = 0     b.  𝑥! − 4𝑥! = 0  c.    𝑥! − 4𝑥 = 0       d.  3𝑎! = 12𝑎𝑏  e.  2𝑥 𝑥 + 1 + 3 𝑥 + 1 = 0  

 10. Write  each  expression  as  a  product  of  expressions  (in  other  words:  factor  

completely.)    

     

11. Consider  the  equation  𝑓 𝑥 = 𝑥! − 4𝑥 + 4  a. Find  a  polynomial  s(x)  such  that  𝑓 𝑥 + 𝑠 𝑥 = 𝑥! − 3𝑥 − 6  b. Find  the  difference  𝑓 𝑥 − 𝑠 𝑥  c. Find  the  product  𝑓(𝑥) ∗ 𝑠(𝑥)  

 12. Consider  the  following  polynomials:    

  2𝑥! − 3𝑥! + 𝑥       2𝑥! + 8𝑥! − 1  a. Find  their  sum.  What  is  the  degree  of  their  sum?  b. Find   their  difference   if   the  second   is  subtracted   from  the   first.  What   is   the  

degree  of  their  difference?  c. Find  their  product.  What  is  the  degree  of  their  product?  

 13. Consider  the  polynomial  2− 3𝑥! + 2𝑥! − 4𝑥 − 𝑥! + 𝑥!  

a. Write  the  polynomial  in  standard  form  (normal  form.)  b. What  is  the  degree  of  the  polynomial?  c. What  is  the  coefficient  of  the  linear  term?  d. What  is  the  coefficient  of  the  constant  term?  e. Find  a  polynomial  that,  when  added  to  this  one  gives  a  sum  of  degree  3.  

 14. Prove  that  each  equation  is  an  identity  by  writing  both  sides  in  standard  form:  

a. 𝑎! + 8 = 𝑎 𝑎! − 2𝑎 + 4 + 2 𝑎! − 2𝑎 + 4  b. (𝑏 − 1)! + 3 𝑏 − 1 = (𝑏 + 2)(𝑏 − 1)  

 15. Factor  each  pair  of  expressions.  

a. 49𝑐! − 16  and  0.49𝑐! − 0.16  b. 121𝑝! − 81𝑝  and  121𝑝! − 81𝑝!  c. 9𝑥! − 16  and  !

!𝑥! − !

!"  

     

 

16. Factor each expression. Use an efficient method. a. 𝑥! + 12𝑥 + 27 d. 4𝑥! − 9 b. 𝑥! − 𝑥 − 12 e. 𝑥! + 13𝑥 + 42 c. 𝑥! − 12𝑥 + 27 f. 𝑥! − 6𝑥 − 720

17. What value of m makes each trinomial a perfect square?

a. 𝑥! +𝑚𝑥 + 16 b. 𝑥! − 20𝑥 +𝑚

18. Use ZPP and factoring to find the solution to each equation.

a. 𝑥! − 5𝑥 = 14 c. 𝑥! − 10𝑥 + 2 = 0 b. 𝑥 𝑥 + 3 = 40 d. 𝑥! + 4𝑥 = 165

19. Factor each of the following completely. You may need to use more than one technique.

a. 𝑥! + 18𝑥 + 81 f. 16𝑥! − 1 b. 𝑥! + 4𝑥 − 21 g. 𝑥! − 14𝑥 + 49 c. 3𝑥! − 21𝑥 h. 75𝑥! − 27𝑥 d. 9𝑥! − 16 i. 𝑥! + 10𝑥! − 24𝑥 e. 5𝑥! + 15𝑥! − 20𝑥 j. 3𝑥! − 6𝑥! − 45𝑥

20. What value of k makes each equation an identity?

21. The following problems ask you to generalize your understanding of perfect square

trinomials to those that are not monic.

22. Solve each polynomial equation. Use completing the square. Check your answers by plugging them back into the original equation.

 

23. Solve the polynomial equation 𝑥! − 12𝑥 − 5 = 8 by completing the square. Explain your steps in writing.

24. Solve each polynomial equation. Use completing the square or any other method. Check

your answers by plugging them back into the original equation.

25. A square piece of sheet metal is 18 inches by 18 inches. You cut a small square from each of its corners to form an open-top square box with a bottom area of 196 square inches.

a. Draw the square sheet. Indicate where to make cuts. b. What size squares should you cut from the corners? Show your computations. c. What is the area of the original square sheet? What is the area of each square that

you cut from the piece of sheet metal?

26. Find the number you would have to add to each of these expressions to make it a perfect square trinomial.

27. The area of the figure below (not drawn to scale) is 26 𝑐𝑚!. Assume all angles that appear to be right are right. Find the value of x.

28. The height of a triangle is 3 cm shorter than its base. If the area of the triangle is 90𝑐𝑚!,

find the base and height of the triangle.