Operations of Rational Expressions. We have studied the concepts of addition, subtraction, multiplication and division of rational numbers in previous classes. Now let us generalize the above for rational expressions. In other words, the product of two rational expression is the product of their numerators divided by the product of their denominators and the resulting expression is then reduced to its lowest form. Thus division of one rational expression by other is equivalent to the product of first and reciprocal of the second expression.
So let's see, negative three times positive eight is gonna be negative And the negative three plus positive eight is positive five. So this actually can be factored into x plus eight times, x plus eight times x minus three. This is what this thing is and if this step looks unfamiliar to you, I encourage you to watch the video on factoring quadratic expressions.
But what's neat about this is now we say, okay, look, to have the same denominator here, we just have to take this fraction and multiply the numerator and denominator by x plus eight. So if we multiply the denominator by x plus eight, then these two denominators are equivalent. But I can't just multiply only the denominator by x plus eight. I also have to multiply the numerator times x plus eight. And so what does this simplify to? This is going to be equal to three over x plus eight times x minus three.
Times x minus three. And then minus, and actually, let me distribute the seven. Minus, remember this negative sign out front, so it's gonna be minus seven. Actually, let me just write it this way first. Minus seven x plus 56 over x plus eight times x minus three. I would try to get to the chase, cut to the chase a little bit faster. But this is hopefully so it helps us understand things.
And so this is going to be equal to. We have our common denominator. X plus eight times x minus three. So it's going to be three. And we can distribute this negative sign.
Minus seven x minus So it's going to be equal to, let's see, we can write negative seven x. But either way works. Finding the least common denominator is the same as finding the least common multiple of 4, 6, and There are a couple of ways to do this.
The first is to list the multiples of each number and determine which multiples they have in common. The least of these numbers will be the least common denominator. The other method is to use prime factorization , the process of finding the prime factors of a number. This is how the method works with numbers. The LCM will contain factors of 2, 3, and 5. Multiply each number the maximum number of times it appears in a single factorization.
In this case, 3 appears once, 5 appears once, and 2 is used twice because it appears twice in the prime factorization of 4. Now that you have found the least common multiple, you can use that number as the least common denominator of the fractions.
Multiply each fraction by the fractional form of 1 that will produce a denominator of In the next example, we show how to find the least common multiple of a rational expression with a monomial in the denominator. Find the least common multiple. Both 5 and 7 appear at most once. For the variables, the most m appears is twice. Compare each original denominator and the new common denominator. Remember that m cannot be 0 because the denominators would be 0.
If possible, simplify by finding common factors in the numerator and denominator. This rational expression is already in simplest form because the numerator and denominator have no factors in common. That took a while, but you got through it. Adding rational expressions can be a lengthy process, but taken one step at a time, it can be done.
Find the prime factorization of each denominator. In this case, it is easier to leave the common multiple in terms of the factors, so you will not multiply it out.
Subtract the numerators and simplify. Otherwise you would be subtracting just the t. The video that follows contains an example of adding rational expressions whose denominators are not alike. The denominators of both expressions contain only monomials. The video that follows contains an example of subtracting rational expressions whose denominators are not alike. The denominators are a trinomial and a binomial.
In the next example, we show how to find a common denominator when there are no common factors in the expressions. Because they have no common factors, the least common multiple, which will become the least common denominator, is the product of these denominators.
Then rewrite the subtraction problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.
In the video that follows, we present an example of adding two rational expression whose denominators are binomials with no common factors. You may need to combine more than two rational expressions. While this may seem pretty straightforward if they all have the same denominator, what happens if they do not?
In the example below, notice how a common denominator is found for three rational expressions. Once that is done, the addition and subtraction of the terms looks the same as earlier, when you were only dealing with two terms. Find the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization.
Rewrite the original problem with the common denominator. Check for simplest form. In the video that follows we present an example of subtracting 3 rational expressions with unlike denominators. One of the terms being subtracted is a number, so the denominator is 1. The LCM becomes the common denominator. Multiply each expression by the equivalent of 1 that will give it the common denominator.
The domain is found by setting the denominators equal to zero. In this last video, we present another example of adding and subtracting three rational expressions with unlike denominators.
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