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# Functions math problem

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The polynomial $$f(x)$$ satisfies $$f(x+1)-f(x)=6x+4.$$ Find the leading coefficient of $$f(x)$$

So far, I'm convinced that f(x) is a*(6x+4)+b, but I'm stuck. I tried 6, but the leading coefficient is not 6.

All help is appreciated! Thanks!

Sep 9, 2020

### 1+0 Answers

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WLOG let f(x) be linear. So f(x)=ax+b. So $$f(x+1)=a(x-1)+b=ax-a+b\rightarrow ax-a+b-ax-b=-a$$. Note that this is constant, so f(x) must be quadratic. Let f(x)=ax^2+bx+c. So $$f(x+1)=a(x+1)^2+b(x+1)+c=ax^2+2ax+bx+a+b+c\rightarrow$$$$f(x+1)-f(x)=2ax+c\implies 2a=6\rightarrow \boxed{a=3}$$

Sep 9, 2020