Finding the LCM of Polynomials: \(x^2 - 8x + 15\) and \(x^2 - 5x + 6\)

To find the Least Common Multiple (LCM) of algebraic expressions, specifically polynomials in this case, we follow a process similar to finding the LCM of numbers. The key steps involve factoring each polynomial into its prime factors and then taking the highest power of each distinct factor present in any of the polynomials.

Step 1: Factor the First Polynomial \(x^2 - 8x + 15\)

We need to factor the quadratic expression \(x^2 - 8x + 15\). We look for two numbers that multiply to the constant term (15) and add up to the coefficient of the middle term (-8). The numbers that satisfy these conditions are -3 and -5 (\((-3) \times (-5) = 15\); \((-3) + (-5) = -8\)).

So, the factored form is:

\(x^2 - 8x + 15 = (x - 3)(x - 5)\)

Step 2: Factor the Second Polynomial \(x^2 - 5x + 6\)

Next, we factor the quadratic expression \(x^2 - 5x + 6\). We look for two numbers that multiply to the constant term (6) and add up to the coefficient of the middle term (-5). The numbers that satisfy these conditions are -2 and -3 (\((-2) \times (-3) = 6\); \((-2) + (-3) = -5\)).

So, the factored form is:

\(x^2 - 5x + 6 = (x - 2)(x - 3)\)

Step 3: Identify the Factors and Their Powers

Now we list the prime factors obtained from the factorisation of each polynomial:

• Factors of \(x^2 - 8x + 15\) are \((x - 3)\) and \((x - 5)\).
• Factors of \(x^2 - 5x + 6\) are \((x - 2)\) and \((x - 3)\).

The distinct factors appearing in either polynomial are \((x - 2)\), \((x - 3)\), and \((x - 5)\). We need to consider the highest power of each distinct factor present in either expression.

• The factor \((x - 2)\) appears only in the second polynomial with a power of 1. The highest power is \((x - 2)^1\).
• The factor \((x - 3)\) appears in both polynomials with a power of 1. The highest power is \((x - 3)^1\).
• The factor \((x - 5)\) appears only in the first polynomial with a power of 1. The highest power is \((x - 5)^1\).

Step 4: Calculate the LCM

The LCM is the product of the highest powers of all the distinct factors:

LCM \(= (x - 2)^1 \times (x - 3)^1 \times (x - 5)^1\)

LCM \(= (x - 2)(x - 3)(x - 5)\)

Summary of Factorisation and LCM

Distinct factors are \((x - 2), (x - 3), (x - 5)\). Highest power for each is 1.

LCM \(= (x - 2)(x - 3)(x - 5)\)

Conclusion

The Least Common Multiple of \(x^2 - 8x + 15\) and \(x^2 - 5x + 6\) is \((x - 2)(x - 3)(x - 5)\).

Revision Table: Key Concepts in Polynomial LCM

Additional Information: Polynomial GCD and LCM Relationship

Similar to integers, there is a relationship between the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) and the LCM of polynomials. For two polynomials \(P(x)\) and \(Q(x)\), their product is equal to the product of their GCD and LCM:

\(P(x) \times Q(x) = \text{GCD}(P(x), Q(x)) \times \text{LCM}(P(x), Q(x))\)

In this case, the factors of \(x^2 - 8x + 15\) are \((x - 3)(x - 5)\) and the factors of \(x^2 - 5x + 6\) are \((x - 2)(x - 3)\).

The common factor is \((x - 3)\), so the GCD \(= (x - 3)\).

The LCM \(= (x - 2)(x - 3)(x - 5)\).

Let's verify the relationship:

• \(P(x) \times Q(x) = ((x - 3)(x - 5)) \times ((x - 2)(x - 3))\)
• \(\text{GCD} \times \text{LCM} = (x - 3) \times ((x - 2)(x - 3)(x - 5))\)

While the expressions are related, note that \((x - 3)^2\) appears in the product \(P(x)Q(x)\), whereas in \(\text{GCD} \times \text{LCM}\), \((x-3)\) is in the GCD and \((x-3)\) is in the LCM, resulting in \((x-3)^2\). This relationship holds true, but requires careful handling of repeated factors and leading coefficients in some contexts (though typically simplified for basic problems like this one).

Understanding both LCM and GCD helps in simplifying algebraic fractions involving these polynomials.