Video Transcript
Find the set of zeros of the
function π of π₯ equals seven π₯ to the sixth power minus 112π₯ to the fourth
power.
Weβre asked to find the set of
zeros of this function π of π₯ which is a polynomial. The zeros or roots of a polynomial
π of π₯ are the values that π₯ equals π such that π of π equals zero. Essentially, they are the values of
the variable π₯ that make the value of the function zero.
To find the zeros of π of π₯ then,
we set the function equal to zero. We then observe that both terms
have a shared factor of π₯ to the fourth power. So we can factor by π₯ to the
fourth power to give π₯ to the fourth power multiplied by seven π₯ squared minus 112
equals zero. Furthermore, as seven is a factor
of 112, we can also factor by seven to give seven π₯ to the fourth power multiplied
by π₯ squared minus 16 equals zero. We now have that the product of two
factors is equal to zero.
It follows that at least one of the
individual factors must be equal to zero. And so we have seven π₯ to the
fourth power equals zero, or π₯ squared minus 16 equals zero. We can now solve these equations
separately to find the zeros of π of π₯.
The first equation can be
simplified by dividing both sides by seven. And then, if π₯ to the fourth power
is equal to zero, it follows that π₯ itself must be equal to zero. To solve the second equation, we
begin by adding 16 to both sides to give π₯ squared equals 16. π₯ is then equal to plus or minus
the square root of 16, which is positive or negative four.
Weβve found all solutions to these
equations, and so weβve found all the zeros of π of π₯. Weβre asked to give the answer as a
set. So we have that the set of zeros of
the function π of π₯ is the set containing the values zero, four, and negative
four.
