Bosco

 What is the coefficient of x in (x^3 + x^2 + x + 1)(7x^6 + 8x^5 - 4x^4 - 10x^3 + 12x^2 - 2x - 8)?    

Every term in the first expression is multiplied by every term in the second expression.    

Determine the terms that, when multiplied together, result in the "x" coefficient product.   

(x³ + x² + x + 1)(7x⁶ + 8x⁵ – 4x⁴ – 10x³ + 12x² – 2x – 8)    gives you   – 2x    

(x³ + x² + x + 1)(7x⁶ + 8x⁵ – 4x⁴ – 10x³ + 12x² – 2x – 8)    gives you   – 8x    

                              add them together for the final value                      – 10x    

The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.    

M = [ (12) + (12 + 16) ] / 2    

M = 40/2    

M = 20    

The median of a trapezoid is the average of its two bases.    

In this problem, the area of the trapezoid is irrelevant.    

_.    

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?    

In order not to have two yellows next to each other,    

the three yellows must occupy squares 1, 3, and 5.    

That leaves squares 2 and 4 to fill, and two colors    

to do it with. There are only four ways.   

Blue Blue     Red Red     Blue Red     Red Blue    

So, the answer is four ways.     

_.    

A baking club wants to form an executive committee. There are $20$ people in the baking club, including Mark. In how many ways can the baking club form an executive committee with $7$ people, including Mark?     

₂₀C₇ = 77,520    

This is called the combination of 20 items taken 7 at a time.    

Sometimes called the shortened form of "20 choose 7" but    

I prefer the first way.   

_.    

Let a1, a2, a3, ..., a10 be an arithmetic series. If a1 + a3 = -2 and a2 + a4 + a6 = 1, then find a1.    

call "d" the amount of addition between successive terms.    

starting with a₁   then  a₂  =  a₁ + d                                              =   a₁ + d     

                                    a₃  =  a₂ + d  =  (a₁ + d) + d                     =   a₁ + 2d    

                                    a₄  =  a₃ + d  =  (a₁ + d + d) + d               =   a₁ + 3d    

                                    a₅  =  a₄ + d  =  (a₁ + d + d + d) + d         =   a₁ + 4d    

                                    a₆  =  a₅ + d  =  (a₁ + d + d + d + d) + d   =   a₁ + 5d    

given       a₁ + a₃  =  – 2    

                a₁ + a₁ + 2d  =  – 2    

                2a₁ + 2d  =  – 2    

                     a₁ + d  =  – 1                                   (eq 1)    

given        a₂ + a₄ + a₆  =  1    

                (a₁ + d) + (a₁ + 3d) + (a₁ + 5d)  =  1    

                3a₁ + 9d  =  1                                       (eq 2)    

multiply (eq 1) by – 9         – 9a₁ – 9d  =  9    

add (eq 2)                             3a₁ + 9d  =  1    

                                           – 6a₁          =  10    

                                                a₁          =  – 10 / 6    

                                                        a₁  =  – 5 / 3    

_.   

Every item in the first term is multiplied with every item in the second term.    

Pick out the items that, when multiplied, result in an "x" coefficient product.    

(x3 + x2 + x + 1) • (x4 – 8x3 + 17x2 – 23x + 14)     gives you + 14x    

(x3 + x2 + x + 1) • (x4 – 8x3 + 17x2 – 23x + 14)     gives you – 23x    

add them together for the final value                                     – 9x    

_.    

A line and a circle intersect at A and B as shown below. Find the distance between A and B.    
The line is x = 4, and the equation of the circle is x^2 + y^2 = 25.    

Substitute 4 for x in the equation of the circle to find the point(s) common to both curves.    

x² + y²  =  25    

       y²  =  25 – x²    

       y²  =  25 – 4²    

       y²  =  9    

       y  =  + 3       —>>  The curves intersect at (+4, +3) and (+4, –3)    

Since x=4 is a straight, vertical line, there is no horizontal component of the distance between A and B.    

Therefore, the distance between + 3 and – 3 is 6 units.    

_.    

Every item in the first term is multiplied with every item in the second term. 

Pick out the items that, when multiplied, result in an "x" coefficient product.  

(x³ + x² + x + 1) • (x⁴ – 8x³ + 17x² – 23x + 14)     gives you + 14x    

(x³ + x² + x + 1) • (x⁴ – 8x³ + 17x² – 23x + 14)     gives you – 23x    

add them together for the final value                                     – 9x    

_.