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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 $f(x) + g(x)$?

2. 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 $f(x) + a\cdot g(x)$?

3. 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 $f(x) + b\cdot g(x)$?

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

5. The degree of the polynomial $p(x)$ is 11, and the degree of the polynomial $q(x)$ is 7. Find all possible degrees of the polynomial $p(x) + q(x).$

 Mar 11, 2021
 #1
avatar+121060 
+1

1.  5x^3 + x^3   = 6x^3

 

2.   a  non-zero constant multiplier doesn't  change thedegree of a polynomial

 

So  the LARGEST  possible  degree  of  f(x) + a * g(x)   = 4  

 

 

cool cool cool

 Mar 11, 2021
 #2
avatar+20 
+1

Thanks!

LookingForAnswers  Mar 11, 2021
 #3
avatar+121060 
+1

3.  Note that if  b =  -1/2     we  have

 

(x^4  - 3x^2  + 2)   - (1/2)(2x^4  -6x^2  + 2x  - 1)   =

 

(x^4  - 3x^2 + 2)  - x^4  + 3x^2  - x   + 1/2  =

 

-x + 2 + 1/2   = 

 

- x  + (5/2)x  =     degree  1

 

 

cool cool cool

 Mar 11, 2021
edited by CPhill  Mar 11, 2021
 #4
avatar+20 
+1

you're a life saver. 

LookingForAnswers  Mar 11, 2021
 #5
avatar+121060 
+1

4.    Degree  of    f(x) * (g(x)   =  x^4  * 2x^4   =   2x^8  =   degree 8

 

cool cool cool

 Mar 11, 2021
 #6
avatar+121060 
+1

5.  Note  that since  p(x)  has degree 11  and q(x)  =  7

 

Then  p(x)  + g(x)   will  have  a degree of   11....in other words.....the degree  of q(x)  is irrelevant in the addition

 

 

cool cool cool

 Mar 11, 2021
 #7
avatar+20 
+1

Ok thanks! That's all I needed!

LookingForAnswers  Mar 11, 2021

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