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1. Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. Let $a$ be a constant. What is the largest possible degree of $f(x) + a\cdot g(x)$?

 

2. There is a polynomial which, when multiplied by $x^2 + 2x + 3$, gives $2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9$. What is that polynomial?

 

3.  What is the coefficient of $x$ in $(x^4 + x^3 + x^2 + x + 1)^4$?

 

4. What is the coefficient of $x^3$ in this expression? \[(x^4 + x^3 + x^2 + x + 1)^4\]

Guest Mar 30, 2018
 #1
avatar+86944 
+1

1. \(f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1 \)

 

Let a  be a constant

What  is the highest pssible degree of

 

\(f(x) + a\cdot g(x) \)

 

 

Multiplying g(x)  by a non-zero constant  will not change the degree of g(x)

As long as "a"  is  not equal to   -1/2,    the degree  of  f(x) + a*g(x)  will  be  4

 

cool cool cool

CPhill  Mar 30, 2018
edited by CPhill  Mar 30, 2018
 #2
avatar+86944 
+1

2. There is a polynomial which, when multiplied by $x^2 + 2x + 3$, gives $2x^5 + 3x^4 + 8x^3 + 8x^2 + 18x + 9$. What is that polynomial?

 

We can find this with division

 

                         2x^3  - x^2   + 4x  + 3

x^2 + 2x + 3  [  2x^5 + 3x^4 + 8x^3 +   8x^2 + 18x   + 9  ]

                        2x^5  + 4x^4 + 6x^3

                       ________________________

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

                                     -x^4  - 2x^3  -  3x^2

                                   ______________________

                                               4x^3  + 11x^2 +18x

                                               4x^3   + 8x^2 + 12x   + 9

                                              _____________________

                                                             3x^2 + 6x  +  9

                                                             3x^2  +6x  +  9

                                                            _____________

 

The polynomial we need is in red

 

 

 

cool cool cool

CPhill  Mar 30, 2018
 #3
avatar+86944 
+1

3.  What is the coefficient of x in (x^4 + x^3 + x^2 + x + 1)^4?

 

 

x^16 + 4 x^15 + 10 x^14 + 20 x^13 + 35 x^12 + 52 x^11 + 68 x^10 + 80 x^9 + 85 x^8 + 80 x^7 + 68 x^6 + 52 x^5 + 35 x^4 + 20 x^3 + 10 x^2 + 4 x + 1

 

The coefficient  is  4

 

 

4.   From above....the  coefficient on the x^3  term  is  20

 

 

cool cool cool

CPhill  Mar 31, 2018

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