Problem 85 - Elementary Algebra

Question

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 10 x^{4}+20 x^{3}+10 x^{2} ; 8 x^{3}+16 x^{2}+8 x $$

Step-by-Step Solution

Answer

The LCM is 40x^2(x + 1)^2.

Step 1 - Factor the first polynomial

First, factor out the greatest common factor from the polynomial 10x^4 + 20x^3 + 10x^2.The GCF is 10x^2, so the polynomial can be factored as:10x^4 + 20x^3 + 10x^2 = 10x^2 (x^2 + 2x + 1)

Step 2 - Prime factorize the first polynomial

Next, factor the expression inside the parentheses.The term (x^2 + 2x + 1) can be further factored as:n(x + 1)^2Thus, the polynomial 10x^4 + 20x^3 + 10x^2 becomes:10x^2 (x + 1)^2

Step 3 - Factor the second polynomial

Now, factor out the greatest common factor from the polynomial 8x^3 + 16x^2 + 8x.The GCF is 8x, so the polynomial can be factored as:8x^3 + 16x^2 + 8x = 8x(x^2 + 2x + 1)

Step 4 - Prime factorize the second polynomial

Next, factor the expression inside the parentheses.The term (x^2 + 2x + 1) can be further factored as:(x +1)^2Thus, the polynomial 8x^3 + 16x^2 + 8x becomes:8x(x + 1)^2

Step 5 - Identify the unique prime factors

Identify all the unique prime factors from the prime factorizations: From the first polynomial: 10x^2 (x + 1)^2Prime factors: 2, 5, x, and (x + 1)From the second polynomial: 8x(x + 1)^2Prime factors: 2, x, and (x + 1)

Step 6 - Determine the least common multiple

To find the least common multiple, take the highest power of each prime factor that appears in the factorizations:Prime factor 2: Highest power = 2^3 (8)Prime factor 5: Highest power = 5Prime factor x: Highest power = x^2Prime factor (x + 1): Highest power = (x + 1)^2Therefore, the least common multiple (LCM) is:8 * 5 * x^2 * (x + 1)^2 = 40x^2(x + 1)^2

Key Concepts

prime factorization

Prime factorization is a method used to break down a number or polynomial into a product of its prime elements. Think of it like finding the building blocks. For numbers, these are prime numbers like 2, 3, 5, etc. For polynomials, it means breaking them down to polynomials which cannot be factored further except by 1 and itself. For instance, when we factorize the polynomial \(10x^4 + 20x^3 + 10x^2\), the first step is to identify the greatest common factor (GCF), which is \(10x^2\). This reduces the polynomial to \(10x^2 (x^2 + 2x + 1)\). Further breaking down \((x^2 + 2x + 1)\) gives \((x + 1)^2\). So, we get prime factors as \(10\), \(x^2\), and \((x + 1)^2\). It's simple but very powerful. Prime factorization can simplify complex problems and is essential in finding the least common multiple (LCM) or greatest common factor (GCF) of polynomials.

greatest common factor

The greatest common factor (GCF) is the highest factor that divides two numbers or polynomials. It simplifies equations and makes calculations easier. To find the GCF of a polynomial, identify the largest coefficients and highest powers of variables common in all terms. Take the polynomials \(10x^4 + 20x^3 + 10x^2\) and \(8x^3 + 16x^2 + 8x\). For the first polynomial, the GCF is \(10x^2\) since 10 and \(x^2\) are common. For the second polynomial, the GCF is \(8x\), since 8 and \(x\) are common. By identifying the GCF, you can rewrite polynomials in a factored form which is crucial for further operations like prime factorization and finding the LCM. Recognizing the GCF saves time and simplifies work, making equations easier to handle.

polynomial factorization

Polynomial factorization involves writing a polynomial as a product of its factors, which are polynomials of lower degrees. It's like breaking a complicated job into smaller, easier tasks. In our example, we started with the polynomials \(10x^4 + 20x^3 + 10x^2\) and \(8x^3 + 16x^2 + 8x\). By using polynomial factorization, we factor out the GCF first: \(10x^2 (x^2 + 2x + 1)\) and \(8x(x^2 + 2x + 1)\). Further factorization of \((x^2 + 2x + 1)\) results in \((x + 1)^2\), reducing the polynomials to \(10x^2 (x + 1)^2\) and \(8x (x + 1)^2\). Factoring polynomials helps in solving equations, simplifying expressions, and finding the LCM by examining the highest powers of all variables and constants involved. It’s a critical skill in algebra and higher mathematics, making complex problems more manageable.