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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 b be a constant. What is the smallest possible degree of the polynomial \(f(x) + b\cdot g(x)?\)

 

 

also if you could explane what degree means in functions that would be great too :)

 

Any help will be great.

THX IN ADVANCE \(:)\)

 

srry for all the changes. 

 Aug 21, 2019
edited by Guest  Aug 21, 2019
edited by Guest  Aug 21, 2019
edited by Guest  Aug 21, 2019
 #1
avatar+5798 
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\(1) ~1 + 5 = 6\\~\\ 2)~\text{The degree of a polynomial $f(x)$ is its highest power of $x$ $\\f(x)+bg(x)$ will be of degree 4}\)

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 Aug 21, 2019
 #2
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Answer 1 was correct but 2 was not. thx for question 1 tho !! laugh

 Aug 21, 2019
 #3
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(2)   If we let  b  = -1/2  we have

 

f(x)  + (-1/2) g(x)   =

 

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   ⇒   degree  1   is the smallest possible degree

 

 

 

cool cool cool 

 Aug 21, 2019
 #4
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+1

Thx CPhill The answer was correct laugh

Guest Aug 21, 2019
 #6
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I belive that the motivation for substituting \(b=-\frac 12\) was to cancel out the \(2x^4\) term. Then, you easily work your way to the correct answer.

 Aug 22, 2019

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