Question Video: Simplifying Rational Expressions Mathematics

Fully simplify ((π‘₯ βˆ’ 3)(π‘₯Β² βˆ’ 6π‘₯ + 9))/((π‘₯ βˆ’ 3)Β³).

02:28

Video Transcript

Fully simplify π‘₯ minus three times π‘₯ squared minus six π‘₯ plus nine over π‘₯ minus three cubed.

Remember, to simplify fractions, we divide the numerator and denominator by the greatest common factor of each. Now, we can see that we have a common factor of π‘₯ minus three. There’s an π‘₯ minus three in the numerator and the denominator. But is that the greatest common factor? Well, to check, we factor any nonfactored expressions. So we’re going to fully factor the expression π‘₯ squared minus six π‘₯ plus nine from the numerator of our fraction. This is a quadratic expression, but there are no common factors in π‘₯ squared, negative six π‘₯, and nine. So that tells us we factor into two parentheses.

We know that, in order to achieve an π‘₯ squared, we need an π‘₯ here and an π‘₯ here. And we need to find two numbers whose product is nine and whose sum is negative six. Well, that must be negative three and negative three, since negative three times negative three is positive nine. But negative three plus negative three is negative six. If we replace π‘₯ squared minus six π‘₯ plus nine with its factored form, our fraction becomes π‘₯ minus three times π‘₯ minus three times π‘₯ minus three over π‘₯ minus three cubed. But of course, it should be quite clear that π‘₯ minus three times π‘₯ minus three times π‘₯ minus three is actually π‘₯ minus three cubed.

Notice now that our numerator and denominator are actually equal. We’re dividing a number by itself. And when we divide a number by itself, we get one. So in simplified form, this fraction is simply one. Now, note that we could’ve approached this slightly differently. Instead of factoring at the start π‘₯ squared minus six π‘₯ plus nine, we could have divided through by a factor of π‘₯ minus three. We eventually saw that this isn’t the greatest common factor, but it’s a good start. We divide the numerator and the denominator by π‘₯ minus three, noting that π‘₯ minus three cubed divided by π‘₯ minus three is π‘₯ minus three squared.

Then we could’ve factored and spotted that we had further common factors of π‘₯ minus three squared. Either method ultimately results in us dividing both the numerator and denominator by the greatest common factor of π‘₯ minus three cubed. And either method results in an answer of one.

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