Bosco

     what is the factored form of this polynomial     

     x^3 + 3x^2 - 13x -15     

            x³ + 3x² – 13x – 15     

           (x + 1) • (x + 5) • (x – 3)     

_.     

Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 1 greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator of the previous fraction by 1. He then multiplies all his fractions together. He has 20 fractions, and their product equals 3. What is the value of the first fraction he wrote?     

  x+1     x+2     x+3     x+4             x+17     x+18     x+19     x+20       3     

 –––  •  –––  •  –––  •  –––  • • • •  ––––  •  ––––  •  ––––  •  ––––  =  ––     

   x        x+1     x+2     x+3             x+16     x+17     x+18     x+19        1     

Now, cancellation . . . .     

  x+1     x+2     x+3     x+4             x+17     x+18     x+19     x+20       3     

 –––  •  –––  •  –––  •  –––  • • • •  ––––  •  ––––  •  ––––  •  ––––  =  ––     

   x        x+1     x+2     x+3             x+16     x+17     x+18     x+19        1     

                                                                                  (x + 20) / (x)  =  3     

                                                                                            x + 20  =  3x     

                                                                                                  2x  =  20     

                                                                                                    x  =  10     

                                                                   11     

                           so the first fraction was  –––     

                                                                   10     

_.     

Emanuel was charged $32 for a 14 2/9 taxi ride. The kilometer is constant. What was the cost per kilometer?     

     32.00 / ¹²⁸/₉   =   32.00 x ⁹/₁₂₈   =   0.25 x 9   =   2.25     

_.     

Which expression is equivalent to 2/7(14x + 3)?   2x + 6/7   4x + 2/21   4x + 6/7   4x + 3     

      2        14x + 3            28x + 6            28x          6     

    –––  •  –––––––    =    –––––––    =    ––––  +  –––     =    4x + ⁶/₇       

      7             1                      7                   7            7     

_.     

Shelly has a beaker that contains 5 and two-thirds fluid ounces of water. She pours out 3 and one-third fluid ounces of water. Which expression can be used to find the number of fluid ounces of water that remain in the beaker? 5 and two-thirds minus 3 and one-third 5 and two-thirds + 3 and one-third 3 and one-third minus 5 and two-thirds 5 and two-thirds minus (negative 3 and one-third)     

No wonder you have trouble solving it.  I can't even read that.  Let's do it this way:     

Shelly has a beaker that contains 5 and two-thirds fluid ounces of water.

She pours out 3 and one-third fluid ounces of water.

Which expression can be used to find the number of fluid ounces of water that remain in the beaker?     

i)      5 and two-thirds minus 3 and one-third     <<——  It's this one.     

ii)     5 and two-thirds + 3 and one-third     

iii)    3 and one-third minus 5 and two-thirds     

iv)    5 and two-thirds minus (negative 3 and one-third)

The functions f and h are defined as follows:     

f(x) = (x + 5)/(x - 7)  and h(x) = (x + 5)(x - 7).     

Explain why the functions f and h are not the same function.     

Because f(x) is a division     

       and h(x) is a multiplication.     

_.     

  sin          opposite / hypotenuse        opposite     

 ––––  =  –––––––––––––––––––  =  ––––––––     

  cos         adjacent / hypotenuse        adjacent     

  So, sine divided by cosine equals tangent.     

_.     

Find the coefficient of y^4 in the expansion of (2y - 5 + y^2)^6.     

The coefficient of y⁴ is –14625     

_.     

Amaretta's birthday is July 27, and her brother Enzo's birthday is September 3. Every year, Amaretta and Enzo celebrate by eating cake every day from Amaretta's birthday through Enzo's birthday (including both birthdays). If they did this for the first time in 2003, how many cake-eating days will they have observed by the end of 2020?     

39 ^days/_year  x  28 years  =  1,092 days to eat cake     

July 27 through September 3 inclusive is   39 days     

2003 through 2020 inclusive is                   28 years     

_.     

Lana collects 20-cent and 50-cent coins. The total value of her coins is currently $9.70. If Lana has 4 more 50-cent coins than 20-cent coins, how many of each type does she have?     

Let F stand for the number of 50¢ coins     

Let T stand for the number of 20¢ coins     

50F + 20T  =  970         (equation 1)     

F  =  T + 4                     (equation 2)     

substitute (T+4) in place of F in eq 1     

50(T+4) + 20T  =  970     

50T + 200 + 20T  =  970     

                    70T  =  770     

                        T  =  11     

                        F  =  T + 4     

                        F  =  15     

check answer     

50F + 20T  =  970     

(50)(15) + (20)(11)  =?  970     

   750     +     220    =?  970     

                     970    =    970     true     

_.     

Laura got 33/35 points on an exam. Mathew got 97% correct on the same exam. What percentage lower or higher, did Mathew get correct on their exam than Laura? Round your answer to the nearest hundredth.     

Matthew got                               97.000%     

Laura got 33/35 which equals   94.286%     

Matthew's score was                   2.714%  higher, rounds to  2.71%     

_.     

¿how can I divide 10 by 3 without get decimal?     

A couple of ways to avoid a decimal.     

Use a fraction:       ¹⁰/₃  =   3 ¹/₃     

Use a remainder:   ¹⁰/₃  =  3 R 1     

_.     

Consider two positive even integers less than 19 (not necessarily distinct). When the sum of these two numbers is added to their product, how many different possible values may result?     

Between 1 to 19 are nine even integers, namely 2, 4, 6, 8, 10, 12, 14, 16, 18.     

We will use two of those integers.  How many pairs of integers could there be.     

It's ₉C₂ = 36.  But wait, the combination process calculates different numbers.     

We have to add another 9, to take doubles into account, for example 4 and 4.     

So, adding 36 and 9, we get 45 possibilities, if my reasoning is correct.     

_.     

-7x^2-23x+10=0     

          –7x² – 23x + 10 = 0     

            7x² + 23x – 10 = 0     

             –23 + sqrt(23² –  (4 • 7 • –10))     

     x  =  ––––––––––––––––––––––––     

                             2 • 7     

             –23 + sqrt(809)     

     x  =  –––––––––––––     

                      14     

             –23 + 28.44293     

     x  =  –––––––––––––     

                      14     

     x = +0.38878    

     x = –3.67449     

_.