# Thread: polynomial and rational zeros

1. ## polynomial and rational zeros

1. express 3/23 as the sum of two unit fractions, 3/23=1/a+1/b

2. consider y=x^2. Determine the poin on the function closest to (3,1)

I know answers to the question 1 (
8 and 184) but I don't know how to solve it.

Can someone teach me how to solve them?

2. $\frac{3}{23}=\frac1a+\frac1b=\frac{a+b}{ab}$, so $a+b=3k, ab=23k$ for some integer k that will cancel on top and bottom. Substitute $b=\frac{23k}{a}$, so $a+\frac{23k}{a}=3$, giving the quadratic $a^2-3ka+23k=0$. The determinant is $D=9k^2-92k$, so the first value for k that makes D a perfect square will give you your answer!

Consider the distance D between the points $(x,x^2),(3,1)$. Let $D^2=(x-3)^2+(x^2-1)^2$. To minimize $D^2$, find its derivative and set to zero. You won't get an integer value, and need a calculator.

Hope this helps!