Video Transcript
Find the set of zeros of the
function 𝑓 of 𝑥 equals 𝑥 times 𝑥 squared minus 81 minus two times 𝑥 squared
minus 81.
To find the zeros of a function, we
set the function equal to zero, which means 𝑥 times 𝑥 squared minus 81 minus two
times 𝑥 squared minus 81 is equal to zero. We can solve this equation for 𝑥
to find the zeros of 𝑓. We can see immediately that all of
the terms on the left-hand side have a common factor of 𝑥 squared minus 81. So we can factorize the left-hand
side with this term. This gives us 𝑥 minus two times 𝑥
squared minus 81 equals zero.
Now notice that 81 is a square
number, nine squared. So, in this second term, we have a
difference of two squares. When we have an expression of the
form 𝑎 squared minus 𝑏 squared, we can factorize this to give 𝑎 minus 𝑏 times 𝑎
plus 𝑏. In our case, this means we can
factorize 𝑥 squared minus 81 to give 𝑥 minus nine times 𝑥 plus nine. We now have a product of binomial
terms, linear in 𝑥. So we cannot factorize any
further. This is a product of three terms,
which is equal to zero. Therefore, at least one of the
terms must itself be equal to zero. Therefore, either 𝑥 minus two
equals zero, 𝑥 minus nine equals zero, or 𝑥 plus nine equals zero.
We can solve the first equation by
adding two to both sides to give 𝑥 equals two, the second equation by adding nine
to both sides to give 𝑥 equals nine, and the third equation by subtracting nine
from both sides to give 𝑥 equals negative nine. Therefore, the set of zeros of 𝑓
is negative nine, two, and nine.
