If the set of zeros of the function 𝑓 of 𝑥 equals 𝑥 squared plus 𝑏 over 𝑥 cubed plus 343 is negative eight and eight, find the value of 𝑏.
Our function looks like this: 𝑥 squared plus 𝑏 over 𝑥 cubed plus 343. And we know where the zeros are. So we want to set this equation equal to zero. When everything on this side of the equation equals zero, what is 𝑏?
First, let’s try to get 𝑏 by itself. To do that, I’ll multiply the left side of the equation by 𝑥 cubed plus 343. If I multiply that value on the left, I need to multiply it by that value on the right. On the left side, those values cancel out, leaving us with 𝑥 squared plus 𝑏. And then zero times anything equals zero. So the right side of our equation remains zero. 𝑥 squared plus 𝑏 equals zero.
Now, it’s time to use this information. We know that the zeros occur when 𝑥 equals negative eight or positive eight. So let’s break this up into two separate equations: negative eight squared plus 𝑏 equals zero and eight squared plus 𝑏 equals zero. Negative eight squared equals 64. 64 plus 𝑏 equals zero. If we subtract 64 from both sides, 𝑏 is equal to negative 64.
We can follow the same process with the second equation. Eight squared equals 64 plus 𝑏 equals zero. And because negative eight squared is the same thing as eight squared, we’ve ended up with the same equation. 𝑏 equals negative 64 in both cases. The only value 𝑏 can be, if the zeros of this function is at negative eight and eight, is negative 64. 𝑏 is equal to negative 64.