Video: Dividing a Polynomial by a Binomial Using Factorization

Simplify (2π‘₯Β² + 5π‘₯ βˆ’ 3)/(π‘₯ + 3).

03:20

Video Transcript

Simplify two π‘₯ squared plus five π‘₯ minus three over π‘₯ plus three.

First, we recall that this fraction line means divide. And so when we simplify, we’re actually saying how can we divide two π‘₯ squared plus five π‘₯ minus three by π‘₯ plus three. Well, once our division problem is written as a fraction, we look to factor where possible. Now, it’s not possible to factor the expression on the denominator. But we can factor the numerator. Let’s look to factor two π‘₯ squared plus five π‘₯ minus three. There are a number of ways of doing this. One way is called the AC method. It’s called the AC method because given a quadratic equation of the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐, we begin by multiplying the value of π‘Ž and 𝑐. In our equation, π‘Ž, which is the coefficient of π‘₯ squared, is two and 𝑐 is negative three. Two multiplied by negative three is negative six.

Our next step, is just like when we factor a quadratic equation where the coefficient of π‘₯ squared is one. We look for two numbers that multiply to make negative six and add to make five. Well, six multiplied by negative one is negative six. And six plus negative one is five. And so we’re going to look to split this middle term up into six π‘₯ and negative one π‘₯. We now write our quadratic as two π‘₯ squared plus six π‘₯ minus one equals three. Now, we’ve not done anything mind-blowing here. We’ve just rewritten our original expression. If we were to now simplify the expression on the right-hand side, that would take us back to the expression on the left.

The next step is to consider the two pairs of terms. We’re going to factor each pair. We see that two π‘₯ squared and six π‘₯ have a highest common factor or a greatest common factor of two π‘₯. And so two π‘₯ squared plus six π‘₯ can be written as two π‘₯ times π‘₯ plus three. Similarly, negative one π‘₯ minus three have a common factor of negative one. So when we factor this expression, we get negative one π‘₯ plus three.

Notice now that each term contains a factor of π‘₯ plus three. And so we can factor by π‘₯ plus three. Two π‘₯ times π‘₯ plus three divided by π‘₯ plus three gives us two π‘₯. Then negative one times π‘₯ plus three divided by π‘₯ plus three gives us negative one. And so we fully factored our quadratic. It’s π‘₯ plus three times two π‘₯ minus one. And this is great because we could now rewrite our fraction. We’ve replaced the quadratic with its factored form. And we see it’s equal to π‘₯ plus three times two π‘₯ minus one all over π‘₯ plus three.

Now that it’s in this form, we can simplify our fraction as we would in numerical fraction by dividing through by any common factors. In this case, we can divide through by π‘₯ plus three. When we do, we see that our expression fully simplifies to two π‘₯ minus one over one or simply two π‘₯ minus one. And so the answer to our question and, in fact, the answer to two π‘₯ squared plus five π‘₯ minus three divided by π‘₯ plus three is two π‘₯ minus one. Now, we did use something called the AC method to factor our quadratic expression. You may be used to using some other method. And that’s absolutely fine as long as you do indeed end up with π‘₯ plus three times two π‘₯ minus one.

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