Video: Finding the Denominator of an Algebraic Fraction in an Equation

Find the denominator of the fraction in this equation: (3π‘₯Β² + 11π‘₯ + 8)/οΌΏ = π‘₯ = 1.

02:46

Video Transcript

Find the denominator of the fraction in this equation: Three π‘₯ squared plus 11π‘₯ plus eight over what is equal to π‘₯ plus one.

Let’s call the denominator of our fraction 𝑦 for now, where 𝑦 is going to be some function of π‘₯. Then we’re going to recall the relationship between fractions and division. A fraction is just another way of writing a division. So what this question is also asking us is, what is the value of 𝑦 such that three π‘₯ squared plus 11π‘₯ plus eight divided by 𝑦 equals π‘₯ plus one? Now, we could rearrange this to make 𝑦 the subject. Or we can quote that if π‘Ž divided by 𝑏 is equal to 𝑐, then π‘Ž divided by 𝑐 must be equal to 𝑏. This makes a lot of sense because if we rearrange each equation, we find that π‘Ž is equal to 𝑏 times 𝑐. We can think of 𝑏 and 𝑐 as a factor pair of π‘Ž. We can therefore say that three π‘₯ squared plus 11π‘₯ plus eight divided by π‘₯ plus one must be equal to 𝑦.

But how do we work out this division on the lef-hand side? Well, we have a number of methods, but factorization is generally the most straightforward. We’re going to look to factor the expression three π‘₯ squared plus 11π‘₯ plus eight. There are a number of ways to do this. One method is kind of observation. It’s a quadratic equation, and there are no common factors apart from one in each of our terms. And so we know we can write it as the product of two binomials. The first term in each binomial must be three π‘₯ and π‘₯ because three π‘₯ times π‘₯ gives us the three π‘₯ squared we need.

And then we need to look for factor pairs of eight bearing in mind that one of these is going to be multiplied by three. And then once that happens, when we add our numbers together, we’re going to get 11. Well, a factor pair we could use is eight and one. And if we multiply three by one, we get three. Then three plus eight is 11. For this to work, both eight and one need to be positive. And so we factored our expression. It’s three π‘₯ plus eight times π‘₯ plus one. And so we can now rewrite our equation as three π‘₯ plus eight times π‘₯ plus one all over π‘₯ plus one equals 𝑦.

Now, our next step since it’s written as a fraction is to simplify like we would any other fraction by dividing through by a common factor. Here, we see we have a common factor of π‘₯ plus one. π‘₯ plus one divided by π‘₯ plus one is simply one. And so we see that 𝑦 is equal to three π‘₯ plus eight? And since we said 𝑦 was the denominator of our fraction, then the denominator is three π‘₯ plus eight. Now, a really quick way to check this answer is to check that the product of π‘₯ plus one and our denominator is indeed equal to three π‘₯ squared plus 11π‘₯ plus eight. And in fact, if we multiply these two binomials, we do indeed get three π‘₯ squared plus 11π‘₯ plus eight.

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