The LCM of 2 x^3+6 x^2 and 6 x^4+4 x^3 is 4 x^3(x+1) |...

Name: Problem 6, Chapter 0: Algebra and Trigonometry
Uploaded: 2021-06-17T15:12:24-08:00
Duration: 4 min 31 s
Channel: Andrew Elmer
Description: The LCM of 2 x^3+6 x^2 and 6 x^4+4 x^3 is 4 x^3(x+1)

Key Concepts

Least Common Multiple (LCM) of Polynomials

The least common multiple of two polynomials is the smallest polynomial that is a multiple of both given polynomials. It is found by factorizing the polynomials into irreducible factors, and then taking each factor to the highest power that appears in either factorization. The LCM is important because it serves as a common denominator in the addition and subtraction of rational expressions and plays a crucial role in simplifying algebraic fractions.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors, which are polynomials of lower degree. This process often involves identifying and factoring out the greatest common monomial factor, grouping terms, or applying special formulas. Factoring simplifies operations such as finding least common multiples and greatest common divisors, and it is a key technique for solving polynomial equations and simplifying expressions.

Greatest Common Divisor (GCD) of Polynomials

The greatest common divisor (or greatest common factor) of two polynomials is the highest-degree polynomial that divides both of the given polynomials without leaving a remainder. It is often used in concert with the LCM, and there is a relationship between the product of the two polynomials, their GCD, and their LCM. Understanding the GCD helps simplify rational expressions and is fundamental in polynomial division and factorization methods.

Transcript

0:00 All right.

00:01 So in this question, we're asked for the lcm of two things.

00:08 Well, first we should see what the lcm stands for.

00:11 The lcm stands for the least common multiple.

00:20 So a little background.

00:22 If i had some numbers that were all relatively prime like three, five, and seven, well, i could multiply those and that would be the smallest number.

00:31 So three times five is five.

00:33 15 and 15 times 7 is 105.

00:42 That if i just multiplied those, then i get 105, and that's the least common number that all of these numbers go into that one.

00:55 So now, what if i had not relatively prime numbers? what if i had three and five and six? well, i know that six is, 3 and 2, and then we got a 5 and a 3.

01:14 And in order to get my least common multiple, i need to multiply a 3 and a 5 and a 2.

01:20 I don't need to multiply a second 3.

01:22 So that would be 3 times 5 is 15 times 2.

01:25 It's 30.

01:29 We could be multiplied 3 and 5 and 2.

01:34 And we think, oh, does 5 go into 30? yeah.

01:38 6 go into 30? yeah.

01:39 5 times 6 is 30, and then 3 also goes into 30.

01:44 So now, if we do that same thing, but just with algebraic expressions, we're going to be looking at 2x to the 3rd plus 6x squared, and 6x to the 4th plus 4x to the 3rd.

02:04 And what we want to know is 4x to the 3rd, x plus 1, is that that least common multiple between us...

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