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# maths; very confused

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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

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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)

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

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Mar 18, 2019