How To Solve LCM in Polynomials

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This educational video provides a comprehensive lesson on finding the Least Common Multiple (LCM) and Highest Common Factor (HCF) of polynomials. The lecture begins by defining LCM as the smallest polynomial divisible by all given polynomials and demonstrates its calculation through factoring, using examples like P(x) = x² - 9 and Q(x) = x² - 3x. The video then introduces the relationship between HCF and LCM, presenting the formula HCF × LCM = Product of the given polynomials, and illustrates this with a worked example. A section on 'Shortcut Tricks' is provided, including rules for determining HCF based on degree, common factors, and division. The video concludes with a practice question asking for the HCF of x² - 9 and x² - 3x, reinforcing the concepts taught.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The video opens with a title slide, "What is LCM of Polynomials?". It defines LCM as the smallest polynomial that can be divided by all given polynomials. Two examples are shown. Example 1 factors P(x) = x² - 9 as (x-3)(x+3) and Q(x) = x² - 3x as x(x-3). The LCM is determined by taking the highest power of each unique factor, resulting in LCM = x(x-3)(x+3). Example 2 follows a similar process for P(x) = x² - 5x + 6 and Q(x) = x² - 3x, yielding LCM = x(x-2)(x-3).

2. 2:00 – 5:00 02:00-05:00

   The video transitions to a comparison between the numeric and polynomial methods for finding HCF. It shows the Euclidean algorithm for numbers 48 and 18, where the HCF is found to be 6. This is contrasted with the polynomial method for P(x) = x³ - 3x² + 2x and Q(x) = x² - 2x + 1, where the HCF is found to be (x-1). The instructor then draws a diagram to illustrate the relationship between the factors of the polynomials, showing how the HCF is the common part and the LCM is the product of all unique factors.

3. 5:00 – 10:00 05:00-10:00

   The instructor uses a diagram to explain the relationship between the factors of the polynomials. He writes the factors of P(x) = x² - 9 as (x-3) and (x+3), and the factors of Q(x) = x² - 3x as x and (x-3). He then shows that the HCF is the common factor (x-3), and the LCM is the product of all unique factors: x(x-3)(x+3). He also demonstrates the relationship between the HCF and LCM using the formula HCF × LCM = Product of the given polynomials, using the example of 20 and 30 to illustrate the concept.

4. 10:00 – 12:50 10:00-12:50

   The video presents a slide titled "Relation Between HCF and LCM" with the formula: HCF × LCM = Product of the given polynomials. An example is worked out using P(x) = x² - 9 and Q(x) = x² - 3x, showing that HCF = (x-3) and LCM = x(x-3)(x+3). The instructor verifies the formula by multiplying HCF and LCM to get the product of the polynomials. The final section, "Shortcut Tricks," lists five rules: 1) HCF cannot have a degree greater than the smallest polynomial, 2) check for common factors like (x+a) or (x-b), 3) use the formula LCM = (Product / HCF), 4) if one polynomial divides the other, the smaller is the HCF and the larger is the LCM, and 5) always verify the answer using HCF × LCM = Product. The video ends with a practice question: "Find the HCF of x² - 9 and x² - 3x."

The video systematically teaches the concepts of LCM and HCF for polynomials. It starts with a clear definition and two worked examples to demonstrate the factoring method. It then draws a powerful analogy to the numeric method, using the Euclidean algorithm to show the parallel process. The core of the lesson is the relationship between HCF and LCM, which is presented as a fundamental formula. This is reinforced with a detailed example and a set of practical shortcut tricks that provide students with efficient problem-solving strategies. The progression from definition to method to formula to shortcuts creates a logical and comprehensive learning path.