Video Transcript
Given that the algebraic fraction 𝑛 of 𝑥 equals eight 𝑥 times 𝑥 plus four over 𝑥 plus 𝑎 simplifies to 𝑛 of 𝑥 equals eight 𝑥, what is the value of 𝑎?
In order to answer this question, we’re going to need to recognize how we simplify an algebraic fraction. To simplify an algebraic fraction, we need to look for shared or common factors in the numerator and denominator of that fraction. When we’ve identified those, we know that we can cancel through by this shared factor, leaving a much-simpler expression. Take the function 𝑛 of 𝑥 which is eight 𝑥 times 𝑥 plus four over 𝑥 plus 𝑎. Since this fraction simplifies to eight 𝑥, it must be equal to eight 𝑥 for all values of 𝑥.
Now, in fact, on the left-hand side, we do have eight 𝑥 already. So, this tells us that we should be able to cancel by dividing through by a common factor with the remaining terms. In turn, this means that the expression 𝑥 plus four must be equal to 𝑥 plus 𝑎 for all values of 𝑥. And, of course, the only way that this can be true is if 𝑎 itself is equal to four. And so, given that the algebraic fraction simplifies to eight 𝑥, we can deduce that the value of 𝑎 is four.
