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Let $f(x) = x^3 + 3x ^2 + 4x - 7$ and $g(x) = -7x^4 + 5x^3 +x^2 - 7$. What is the coefficient of $x^3$ in the sum $f(x) + g(x)$?

 

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 $f(x) + a•g(x)$?

 

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

 

Suppose $f$ is a polynomial such that $f(0) = 47$, $f(1) = 32$, $f(2) = -13$, and $f(3)=16$. What is the sum of the coefficients of $f$?

 

Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. What is the degree of $f(x) • g(x)$?

 

Find $t$ if the expansion of the product of $x^3 - 4x^2 + 2x - 5$ and $x^2 + tx - 7$ has no $x^2$ term.

 

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?

 

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

 

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

Guest Feb 17, 2018
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4+0 Answers

 #1
avatar+86649 
+1

Let $f(x) = x^3 + 3x ^2 + 4x - 7$ and $g(x) = -7x^4 + 5x^3 +x^2 - 7$. What is the coefficient of $x^3$ in the sum $f(x) + g(x)$?

 

The x^3 coefficient will have a sum of  1 + 5  = 6 

 

 

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 $f(x) + a•g(x)$?

 

Depends upon the value of "a"

 

Note that if   a  = -1/2, the sum of the first two terms in each polynomial is 0....and the sum  of the second two terms in each polynomial is also 0....so the sum will produce a degree of 1.....any other value of "a" will produce a 4th degree polynomial

 

 

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

 

Just like the last, if "b"  is -1/2,  the smallest possible degree will be 1 

 

 

 

Suppose $f$ is a polynomial such that $f(0) = 47$, $f(1) = 32$, $f(2) = -13$, and $f(3)=16$. What is the sum of the coefficients of $f$?

 

Guess that we have a third degree polynomial of the form  ax^3 + bx^2 + cx + d

If  f(0)  =  47, then d =  47

 

And we have this system

 

a + b + c + 47  = 32

8a + 4b + 2c + 47  = -13

27a + 9b + 3c + 47  = 16

 

The solution to this is    a = 52/3, b = -67, c = 104/3, d = 47

 

And the sum of these is    32 

 

 

cool cool cool

CPhill  Feb 18, 2018
edited by CPhill  Feb 18, 2018
 #2
avatar+86649 
+1

Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. What is the degree of $f(x) • g(x)$?

 

The degree will  be the result of the product of  x^4 * 2x^4  =  2x^8  ⇒   degree 8   

 

 

 

Find $t$ if the expansion of the product of $x^3 - 4x^2 + 2x - 5$ and $x^2 + tx - 7$ has no $x^2$ term.

 

The  x^2 term in the product  will be given by

 

( 28 + 2t - 5)      so   

 

28 + 2t - 5  = 0

 

2t  + 23  = 0

 

2t  = -23    ⇒    t   =  -23/2 

 

 

cool cool cool

CPhill  Feb 18, 2018
edited by CPhill  Feb 18, 2018
 #3
avatar+86649 
+1

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 polynomial 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

                         _____________________________

                                  - 1x^4  + 2x^3 + 8x^2

                                  - 1x^4  -  2x^3  - 3x^2

                                  __________________

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

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

                                                _________________

                                                            3x^2  + 6x  + 9

                                                            3x^2  + 6x  + 9

                                                            _____________

                                                                                 0

 

 

cool cool cool

CPhill  Feb 18, 2018
 #4
avatar+86649 
+1

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

 

Here's the expansion

 

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  answer to the last question  is   20

 

 

cool cool cool

CPhill  Feb 18, 2018

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