We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
36
1
avatar

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

 Mar 17, 2019
 #1
avatar+4772 
+1

1)

\(a\cdot g(x) \text{ will always have degree 4 for }a\neq 0\\ \text{assuming no cancellation the sum will have degree 4 as well}\\ \text{this is the largest degree the sum can have}\)

 

2)

\(\text{let }b=-\dfrac 1 2\\ f(x)+b\cdot g(x) = (1-1)x^4 + (-3+3)x^2 -x+\left(2+\dfrac 1 2\right)=\\ -x+\dfrac 5 2 \text{ which has degree 1}\)

.
 Mar 18, 2019

6 Online Users