Video Transcript
Simplify 𝑛 of 𝑥 equals three 𝑥 squared minus two 𝑥 over two 𝑥 plus eight minus five 𝑥 minus two over three 𝑥.
Here, we have a function 𝑛 of 𝑥, which is the difference of two algebraic fractions. In order to simplify 𝑛 of 𝑥, we need to combine these two fractions into a single fraction. We notice, first of all, that the denominators of these two fractions are not the same. And so we recall that in order to add or subtract any two fractions, we must first find a common denominator. Let’s use the numeric example of three-sevenths minus one-quarter to recap what we need to do.
Well, first, we need to rewrite both fractions with a common denominator, which will be the lowest common multiple of the individual denominators. In this case, the lowest common multiple of seven and four is 28 because they are coprime. We then need to multiply the numerator of each fraction by whatever we’ve multiplied the denominator by which in this case will be the other denominator. So in the case of three-sevenths, we multiply both the numerator and denominator by the other denominator of four. And in the case of one-quarter, we multiply both the numerator and denominator by the other denominator, which is seven. We can perform the subtraction by subtracting the numerators, and we have our answer.
Let’s now see how we can apply this same process to our example which is algebraic. First, we need to find the lowest common multiple of the two denominators, which will be the common denominator we’ll use to perform the subtraction. Well, as the expressions two 𝑥 plus eight and three 𝑥 have no common factors other than one, their lowest common multiple will be their product, in the same way that the lowest common multiple of seven and four, which are coprime, was their product. We want to express each part of 𝑛 of 𝑥 then as an equivalent fraction, with the denominator of three 𝑥 multiplied by two 𝑥 plus eight, which means we need to multiply the numerator and denominator of each fraction by the other denominator.
For the first fraction, we multiply both the numerator and denominator by three 𝑥. And for the second, we multiply both the numerator and denominator by two 𝑥 plus eight. So we have three 𝑥 squared minus two 𝑥 multiplied by three 𝑥 all over two 𝑥 plus eight multiplied by three 𝑥 minus five 𝑥 minus two multiplied by two 𝑥 plus eight over two 𝑥 plus eight multiplied by three 𝑥. As we now have a common denominator for the two fractions, we can perform the subtraction. This gives three 𝑥 multiplied by three 𝑥 squared minus two 𝑥 minus five 𝑥 minus two multiplied by two 𝑥 plus eight all over three 𝑥 multiplied by two 𝑥 plus eight.
Next, we need to simplify by distributing the parentheses in the numerator. The first set is relatively straightforward. Three 𝑥 multiplied by three 𝑥 squared is nine 𝑥 cubed. And three 𝑥 multiplied by negative two 𝑥 is negative six 𝑥 squared. We need to be a little bit more careful with the second set of parentheses because, firstly, we are distributing parentheses that each contain two terms and secondly we are then subtracting the entirety of this expression.
Let’s begin by just distributing the parentheses. Using the FOIL method, we have 10𝑥 squared plus 40𝑥 minus four 𝑥 minus 16, which simplifies to 10𝑥 squared plus 36𝑥 minus 16. Remember, though, that we are subtracting this entire expression. So we must make sure that we include parentheses around it when we substitute it back into the numerator. We can then distribute this negative sign over the parentheses. Multiplying each term by negative one will change its sign. So we can remove the parentheses but change the sign of each term. So we now have negative 10𝑥 squared minus 36𝑥 plus 16.
The final step is to simplify this expression by collecting any like terms. The only like terms are the negative six 𝑥 squared and negative 10𝑥 squared in the numerator, which combine to give negative 16𝑥 squared. In the denominator, we note that the expression two 𝑥 plus eight has a common factor of two in both terms, which we can factor by. This means we now have six 𝑥 outside the parentheses. And inside the parentheses, we have 𝑥 plus four. This expression can’t be simplified any further as there are no common factors other than one in the numerator and denominator.
So we found that the simplified form of 𝑛 of 𝑥 is nine 𝑥 cubed minus 16𝑥 squared minus 36𝑥 plus 16 all over six 𝑥 multiplied by 𝑥 plus four.
