Identify the least common denominator of each group of...

Identify the least common denominator of each group of rational expression, and rewrite each as an equivalent rational expression with the LCD as its denominator.

$\frac{t}{t^{2}-4 t-21}, \frac{2}{t+3}, \frac{4}{t^{2}-49}$

Key Concepts

Least Common Denominator (LCD)

The least common denominator is the smallest expression that contains each factor of the denominators in a group of rational expressions. Once identified, it can be used to rewrite each rational expression so that they share a common denominator, facilitating operations such as addition or subtraction.

Equivalent Rational Expressions

Rewriting a rational expression with a new, common denominator involves multiplying the numerator and denominator by appropriate factors. This process creates an equivalent expression that maintains the original value while sharing the least common denominator with other expressions.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are fundamental in algebra and require careful manipulation, especially when adding, subtracting, or comparing fractions by ensuring they have a common denominator.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors, which may be linear or of higher degree. This process is essential when simplifying rational expressions and finding the least common denominator, as it reveals the irreducible factors present in each denominator.

Transcript

00:01 We're trying to find the least common denominator of the three rational expressions we are given.

00:05 And to do that, let's start by trying to factor the three denominator.

00:08 So t plus three doesn't look like a good me factor anymore.

00:11 So let's just leave it as it is.

00:13 As for t squared minus 4, 49, we can factor into binomial using what we know about binomial rules.

00:20 So we got t minus 7 and t plus 7 in the other binomial.

00:26 Because we know t squared is a square of t and 49 is the square of seven.

00:37 And in the third denominator becomes also can be factored into a binomial.

00:43 T and one of the, and one sign is positive and one side is negative because both signs of the expanded expression are negative, both the middle and last terms.

00:54 So we're looking at, now we're looking at pairs of the factors of 21.

01:00 And one possible one that could work is seven.

01:02 So seven times three is 21, and negative seven plus three is equal to negative four in the middle term.

01:10 So that seems to be, it seems to work here.

01:14 And now what we want to do is for the lcd, we combine, we multiply all the unique factors between these three denominators together.

01:25 So we get t plus 3, t plus 7, and t minus 7.

01:33 And so this is our new least common denominator.

01:37 And now let's try converting these three fractions.

01:40 So t divided by t plus 3...