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Dividing Radicals and Rationalizing the Denominator - Concept
Long division can be used to divide a polynomial by another polynomial, in this case a binomial of lower degree. When dividing polynomials, we set up the problem the same way as any long division problem, but are careful of terms with zero coefficients. For example, in the polynomial x^3 + 3x + 1, x^2 has a coefficient of zero and needs to be included as x^3+ 0x^2+3x+1in the division problem.
One thing to remember about simplifying radical expressions is thou shall not have a radical in the denominator. What I'm talking about is you don't want to have any square roots in the bottom of the fraction. In order to get it out of the bottom of the fraction, you're going to have to use a bunch of techniques.
First thing, if you're given a fraction that has a square root in the bottom, if you don't want to reduce the fraction first that's a possibility. Another thing you might want to try doing is looking for the perfect square factors and reducing it like you guys have been doing with radical expressions all along.
A couple of things to keep in mind also when you see fractions. The square root of 3 plus square root of 7 is not the same thing as the square root of 3+7. That's a really important distinction. That would be true for multiplying square root of 3 times square root of 7 is equal to the square root of 3 times 7. Don't get that stuff confused in your head. So when we're looking at these sums or differences of radical expressions that have different radicands, we're going to be coming across what we call conjugates.
Conjugates look like this. There are two different sums in differences that have the same two terms like I have root 3 plus root 8 and root 3 minus root 8. These are called conjugates and there are some really cool properties that come out when you're multiplying conjugates. If you multiply two conjugates, your result is always an integer or a whole or a whole number. That's a good thing when you're trying to get square roots out of the bottom of a fraction.
So putting it all together, we have a process called rationalising the denominator. If you're given a fraction that has a square root in the denominator, you rationalise the denominator by multiplying the numerator and denominator by the conjugate of the denominator. That'll make a lot more sense when you start looking at examples but again, most important thing to remember is that you never want to leave a radical; expression or that means a square root in the bottom of a fraction. Always always always rationalise by multiplying by the conjugate of the denominator.