CPhill

Very nice....!!!!

-x^2 + 2x - 4       let us complete the square....factor out a "-1"  and we have

- ( x^2 - 2x  + 4)        take 1/2 of -2  = -1, square it  = 1...add and subtract it inside the parentheses

- ( x^2 - 2x + 1  + 4  - 1)       factor the first three terms, simplify the rest

- [ ( x - 1)^2   + 3 )       apply the ngative back across the terms

- ( x - 1)^2  -3 

"a"  = -1

"b"  = -1

"c"  = -3

So

a + b + c  =    -1  +  -1  + -3  =   -5

2√3  is the "simplified" version

The "unsimplified" version  is √12

If $p(x) = x^4 - 3x + 2$, then find the coefficient of the $x^3$ term in the polynomial $(p(x))^3$.

[ p(x] ^2  =   (x^4 - 3x + 2) ( x^4 - 3x + 2)  =

x^8 - 3x^5 + 2x^4

                  -3x^4  + 9x^2 - 6x

                 +2x^4              -6x    + 4

_____________________________

x^8  - 3x^5  + x^4  + 9x^2 - 12x  + 4

And

[p(x)]^2  * [p(x) ] =  [p(x)] ^3  =

 [x^8  - 3x^5  + x^4  + 9x^2 - 12x  + 4]  * [ x^4 - 3x + 2 ]

The x^3  term  will be    9x^2 * -3x   =   - 27x^3

So.....the coefficient is   -27

If $f(x)$ and $g(x)$ are polynomials such that $f(x) + g(x) = -2 + x,$ then what is $g(x)$ if $f(x) = x^3 - 2x - 2$?

We have that

f(x)  + g(x)  = x - 2     and      so

[ x^3 - 2x - 2 ]  +  g(x)   = x - 2      subtract  x^3 from both sides...add  2x, 2 to both sides

g(x)  = -x^3 + 3x 

 -4/x+1 = -1/x+5      cross  multiply

-4( x + 5)  = -1 ( x + 1)     simplify

-4x - 20  = -1x  - 1      add 1x, 20  to both sides

-3x = 19     divide both sides by -3

x  = -19 / 3

Check solution

-4/[ -19/3 + 1 ]  = -1 / [ -19/3 + 5]

-4/ [ -19/3 + 3/3]  = -1 / [ -19/3 + 15/3]

-4/ [ -16/3]  = -1 / [ -4/3]

-4*3 / -16  = -1 * 3 / -4

-12/ - 16  = -3 / -4

3/4  = 3/4

(x diamond y )  = ( 9 diamond 1)  =  √[9*1] / l 9 - 1 l   = √9 / l 8 l   =  3/8

(x diamond y )  = (4 diamond 3/8)  =  

√[ 4 * (3/8) ] / [ 4 - 3/8 l  = 

√ [12/8 ] / l 32/8 - 3/8 l = 

√ [3/2] / l 29/8 l  =

(8/29)* √ 1.5   ≈   0.3 

Note that we  have an adjacent side to the 31°  angle...and we are looking for the opposite side

So...the tangent seems the natural choice, here....so...

tan (31°)  = x  / 22         multiply both sides by 22

22 *  tan (31°)  = x  ≈  13.2  ft

Here's another way using similar triangles

Call the center of the larger circle, C   and the center  of the smaller circle, D

And call the point formed by the tangent and the radius at the  "top" of the smaller circle, E

Note that the vertical angles formed by the intersection of the tangent lines are equal...and if we connect the  centers of each circle, these  angles will be bisected and equal

So...we have triangles ABC  and  EBD, angle BAC  = angle BED

And angles ABC and  EBD  are also equal

Therefore.by  AA congruency...triangle  ABC  is similar to triangle EBD

Then AC / ED  = BC/BD

Then   24 / 18  = BC /BD

Then  4/3  = BC/ BD

Then CD has 7 equal parts

And each equal part is  70/7   = 10

So...BC  is 4 of these   = 40

So...by the Pythagorean Theorem, √ [ BC^2  - AC^2 ] = BC

So....√[40^2  - 24^2 ]  = √1024   = 32 cm   = BC

The only way that  f(x)  + g(x)   is of degree 3  is if the  g(x)  itself is of degree 5   with the first two terms (written in order of descending powers) being -2x^5 + x^4  and the third term being ax^3....where a is a constant.... [there also might be  other terms with  degrees < 3 ]