Video Transcript
Find the set of zeros of the
function 𝑓 of 𝑥 equals 𝑥 cubed minus four 𝑥 squared minus 25𝑥 plus 100 equals
zero, where all three zeros take integer values.
To find the zeros of a function, we
set the function equal to zero. So we have 𝑥 cubed minus four 𝑥
squared minus 25𝑥 plus 100 equals zero. We are given that all three zeros
of the function take integer values. So we may be able to factorize the
polynomial via the grouping method. In the first two terms, we have a
common factor of 𝑥 squared, which we can factorize with 𝑥 minus four. We also have a common factor of
negative 25 in the last two terms. And this also factorizes with the
term 𝑥 minus four. Therefore, we have a common factor
of 𝑥 minus four between the first two terms and the last two terms. We can therefore factorize the
polynomial to give 𝑥 minus four times 𝑥 squared minus 25. 25 is a square number, five
squared. So, in this second term, we have a
difference of two squares.
Recall that when we have an
expression in the form 𝑎 squared minus 𝑏 squared, it can be factorized as 𝑎 minus
𝑏 times 𝑎 plus 𝑏. We can therefore factorize 𝑥
squared minus 25 as 𝑥 minus five times 𝑥 plus five. We now have a product of three
binomials linear in 𝑥. So we cannot factorize any
further. Since we have a product of three
terms equal to zero, at least one of them must be equal to zero itself. Therefore, either 𝑥 minus four
equals zero, 𝑥 minus five equals zero, or 𝑥 plus five equals zero. And we can solve these three
equations for 𝑥 to give 𝑥 equals four, 𝑥 equals five, and 𝑥 equals negative
five. Therefore, the set of zeros of 𝑓
is negative five, four, and five.
