# Video: Simplifying a Quotient of Two Rational Functions

Given that πβ(π₯) = π₯ + (16/(π₯ β 8)), πβ(π₯) = 9π₯ + (144/(π₯ β 8)), and π(π₯) = πβ(π₯) Γ· πβ(π₯), determine π(π₯) in its simplest form.

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### Video Transcript

Given that π one of π₯ equals π₯ plus 16 over eight minus π₯, π two of π₯ equals nine π₯ plus 144 over eight minus π₯, and π of π₯ equals π one π₯ divided by π two π₯, determine π of π₯ in its simplest form.

Weβre going to be working out π of π₯, which is the quotient of π one of π₯ and π two of π₯. So letβs divide π₯ plus 16 over π₯ minus eight β remember thatβs π one β by π two, which is known as π₯ plus 144 over π₯ minus eight. Before we do that, weβre going to need to simplify the expressions in each of these set of parentheses. By adding a denominator of one to π₯ and nine π₯, we see that we can add the fractions in each expression. Weβll create a common denominator in this first expression by multiplying the first term by π₯ minus eight over π₯ minus eight. That gives us π₯ times π₯ minus eight over π₯ minus eight plus 16 over π₯ minus eight. We will multiply nine π₯ over one by π₯ minus eight over π₯ minus eight too. That gives us nine π₯ times π₯ minus eight over π₯ minus eight plus 144 over π₯ minus eight.

And then, we add the numerators. Our first expression becomes π₯ times π₯ minus eight plus 16 all over π₯ minus eight. And our second expression becomes nine π₯ times π₯ minus eight plus 144 over π₯ minus eight. Then, we recall that to divide by a fraction, we multiply by the reciprocal of that same fraction. In this case, we multiply by π₯ minus eight over nine π₯ times π₯ minus eight plus 144. And then if we look carefully, we see we can cross cancel. We can divide by π₯ minus eight. We then multiply the numerator of the first fraction by the numerator of the second and repeat that process for the denominator. So we get π₯ times π₯ minus eight plus 16 times one, which is just that our original expression, over nine π₯ times π₯ minus eight plus 144.

Next, we distribute our parentheses. We multiply π₯ by π₯ and π₯ by negative eight. And we get π₯ squared minus eight π₯. Similarly, we multiply nine π₯ by π₯ and nine π₯ by negative eight. So we end up with π₯ squared minus eight π₯ plus 16 over nine π₯ squared minus 72π₯ plus 144. And in fact, it might look like weβre finished. But we need to find π of π₯ in its simplest form. So here, we notice that we have a common factor in the denominator. That common factor is nine. And if we factor nine out of the denominator, we get nine times π₯ squared minus eight π₯ plus 16. Now, notice we have a common factor. We can divide both the numerator and the denominator by π₯ squared minus eight π₯ plus 16. And so, weβre left with one-ninth.

π of π₯ equals one-ninth.