Video Transcript
Simplify the function 𝑛 of 𝑥 equals 𝑥 squared minus 121 over 𝑥 to the fourth power minus 11𝑥 cubed and determine its domain.
Now when looking to simplify a rational function, that is a function that’s the quotient of a pair of polynomials, we need to determine the domain before we simplify. So let’s begin by finding the domain of 𝑛 of 𝑥. We recall that the domain of a function is the set of possible inputs to that function. And when dealing with a rational function, we need to be really careful with our denominator. We need to make sure we exclude any values of 𝑥 that make the denominator equal to zero.
So to find the values of 𝑥 that we’re going to disregard from the domain of the function, we’re going to find the values of 𝑥 that do make the denominator equal to zero. In other words, which values of 𝑥 satisfy the equation 𝑥 to the fourth power minus 11𝑥 cubed equals zero? To solve this equation, we need to factor the left-hand side. And if we inspect the terms on the left-hand side, we notice that they have a shared factor of 𝑥 cubed. So 𝑥 to the fourth power minus 11𝑥 cubed can be written as 𝑥 cubed times 𝑥 minus 11. This, of course, is equal to zero. And so, for the product of two expressions to be equal to zero, we know that either one or other of those expressions must itself be equal to zero. In other words, 𝑥 cubed is equal to zero or 𝑥 minus 11 equals zero.
The solution to this first equation is simply 𝑥 equals zero. If 𝑥 is zero, 𝑥 cubed must also be zero. Then we solve the second equation by adding 11 to both sides, giving us 𝑥 equals 11. Otherwise, since the function is the quotient of a pair of polynomials and the domain of polynomial functions is the set of real numbers, we know that the domain of 𝑛 of 𝑥 is the set of real numbers not including these values of 𝑥, not including the set containing the elements zero, 11.
So now we have the domain of our function, let’s see how to simplify it. To simplify a rational expression, we need to look for common factors in the numerator and denominator. And the quickest way to spot these is to factor both parts. Now, in fact, we already factored the denominator. And we got 𝑥 cubed times 𝑥 minus 11. So how do we factor the numerator, 𝑥 squared minus 121? Well, an important fact here is to notice that 121 is a square number; it’s 11 squared. And so our numerator can be represented using the difference of squares. It can be written as 𝑥 plus 11 times 𝑥 minus 11. And this means we can rewrite 𝑛 of 𝑥 as 𝑥 plus 11 times 𝑥 minus 11 over 𝑥 cubed times 𝑥 minus 11.
And now we might notice we have a common factor of 𝑥 minus 11. 𝑥 minus 11 divided by 𝑥 minus 11 will be equal to one, as long as 𝑥 is not equal to 11. But remember, we said that 𝑥 equals 11 is excluded from the domain of our function. So 𝑥 minus 11 divided by 𝑥 minus 11 will never give us zero divided by zero, which is undefined. And so we’re left with 𝑥 plus 11 on the numerator and 𝑥 cubed on the denominator.
And so we’ve completed the question. The function 𝑛 of 𝑥 simplifies to 𝑥 plus 11 over 𝑥 cubed, and the domain of this function is the set of real numbers not including the set containing the elements zero and 11.
