Understanding the Least Common Multiple (LCM) of Polynomials

To find the Least Common Multiple (LCM) of two or more polynomials, we first need to factorize each polynomial completely into its prime factors. Then, the LCM is the product of the highest powers of all the distinct factors that appear in any of the polynomials.

Step-by-Step Factorization

Factorizing the First Polynomial: \(8x^3 + 80x^2 + 200x\)

• Identify the common factor: Notice that all terms have a factor of \(8x\).
• Factor out \(8x\): \(8x(x^2 + 10x + 25)\)
• Factor the quadratic expression \(x^2 + 10x + 25\): This is a perfect square trinomial of the form \((a+b)^2 = a^2 + 2ab + b^2\), where \(a=x\) and \(b=5\).
• So, \(x^2 + 10x + 25 = (x+5)^2\).
• The first polynomial factored is: \(8x(x+5)^2\).

Factorizing the Second Polynomial: \(4x^4 + 16x^3 - 20x^2\)

• Identify the common factor: Notice that all terms have a factor of \(4x^2\).
• Factor out \(4x^2\): \(4x^2(x^2 + 4x - 5)\)
• Factor the quadratic expression \(x^2 + 4x - 5\): We look for two numbers that multiply to \(-5\) and add up to \(4\). These numbers are \(5\) and \(-1\).
• So, \(x^2 + 4x - 5 = (x+5)(x-1)\).
• The second polynomial factored is: \(4x^2(x+5)(x-1)\).

Finding the LCM

Now we have the factored forms of both polynomials:

• Polynomial 1: \(8x(x+5)^2\)
• Polynomial 2: \(4x^2(x+5)(x-1)\)

To find the LCM, we take the highest power of each distinct factor present in either polynomial.

• Numerical factors: We have \(8\) and \(4\). The LCM of \(8\) and \(4\) is \(8\).
• Factor \(x\): In the first polynomial, the power of \(x\) is \(x^1\). In the second polynomial, the power of \(x\) is \(x^2\). The highest power is \(x^2\).
• Factor \((x+5)\): In the first polynomial, the power of \((x+5)\) is \((x+5)^2\). In the second polynomial, the power of \((x+5)\) is \((x+5)^1\). The highest power is \((x+5)^2\).
• Factor \((x-1)\): In the first polynomial, \((x-1)\) is not present (power is \(0\)). In the second polynomial, the power of \((x-1)\) is \((x-1)^1\). The highest power is \((x-1)^1\).

Multiplying the highest powers of all distinct factors:

LCM = (\(\text{LCM of } 8 \text{ and } 4\)) \(\times\) (\(\text{Highest power of } x\)) \(\times\) (\(\text{Highest power of } (x+5)\)) \(\times\) (\(\text{Highest power of } (x-1)\))

LCM = \(8 \times x^2 \times (x+5)^2 \times (x-1)\)

So, the LCM is \(8x^2(x+5)^2(x-1)\).

The LCM is the product of these highest powers: \(8 \times x^2 \times (x+5)^2 \times (x-1) = 8x^2(x+5)^2(x-1)\).

Summary of Findings

The LCM of \((8x^3 + 80x^2 + 200x)\) and \((4x^4 + 16x^3 - 20x^2)\) is \(8x^2(x+5)^2(x-1)\).

Revision Table: Key Steps for Polynomial LCM

Additional Information: GCF and LCM Relationship

The Greatest Common Factor (GCF) and Least Common Multiple (LCM) of polynomials are related. The GCF is the product of the lowest powers of the common factors. Let's find the GCF for the given polynomials:

• Polynomial 1: \(8x(x+5)^2 = (2^3)x^1(x+5)^2\)
• Polynomial 2: \(4x^2(x+5)(x-1) = (2^2)x^2(x+5)^1(x-1)^1\)

Common factors are \(2\), \(x\), and \((x+5)\).

• Lowest power of 2: \(2^2 = 4\)
• Lowest power of \(x\): \(x^1\)
• Lowest power of \((x+5)\): \((x+5)^1\)

GCF = \(4x(x+5)\).

For any two polynomials \(P(x)\) and \(Q(x)\), the product of their GCF and LCM is equal to the product of the polynomials themselves (up to a constant factor):
\(GCF(P(x), Q(x)) \times LCM(P(x), Q(x)) = P(x) \times Q(x)\)

This relationship holds true and is a useful property when dealing with polynomial factorization and multiples.