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

1b.) Using the same equations as before, let b be a constant. what is the smallest possible degree of the polynomial f(x)+b*g(x)?

2) 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?

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

4) 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.

5) 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?

Im sorry for the long list of questions. Thank you!!!

 Apr 4, 2020
 #1
avatar+111329 
+1

Here's a few, BigChungus

 

1a)   Multiplying  a polynomial by a  constant  doesnot change its degree

 

So....adding two 4th power polynomial together  still produces a 4th power  polynomial

 

 

1b )  Let  b  = -1/2

 

So    (1/2) g(x)  produces     -x^4  - x   + 1/2

 

Adding  this to  f(x)  will produce    -3x^2  - x + 5/2

 

So....the smallest  that   f + b*g    can be is  degree  2

 

 

cool cool cool

 Apr 4, 2020
 #2
avatar+111329 
+1

2) 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?

 

If f(0) =  47....then  the  constant term  must  be  47

 

And if  f(1)  = 32.....then the  sum of the coefficients and the  constant term  = 32

 

Therefore

 

sum of coefficients  + constant term =  32

 

sum of coefficients  + 47 =  32     subtract  47  from both sides

 

sum of coefficients =   -15

 

cool cool cool

 Apr 4, 2020
 #3
avatar+158 
0

Thank you for your hard work!!! Unfortunately, the site just started maintenance and its likely my class will end. Sorry for causing trouble

 Apr 4, 2020
 #4
avatar+111329 
+1

3)

 

f(x) * g(x)  =  degree  4  * degree 4  =   degree 8   

 

 

cool cool cool

 Apr 4, 2020

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