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Chapter 6: Problem 25

Find the least common multiple of each pair of polynomials. $$ 8 x^{2}, 12 x^{5} $$

Short Answer

Expert verified
The LCM of 8x^2 and 12x^5 is 24x^5.

Step by step solution

01

Factorize Each Polynomial

Factorize the given polynomials into their prime factors. For the polynomial 8x^2, the factorization is \(8x^2 = 2^3 \times x^2\). For the polynomial 12x^5, the factorization is \(12x^5 = 2^2 \times 3 \times x^5\).
02

Identify the Highest Powers of Each Factor

For each distinct factor, determine the highest power that appears in the factorization of either polynomial. For the factor 2, the highest power is 3 (from 2^3 in 8x^2). For the factor 3, the highest power is 1 (from 3 in 12x^5). For the factor x, the highest power is 5 (from x^5 in 12x^5).
03

Multiply the Highest Powers Together

Multiply together the highest powers of each factor identified in Step 2. This gives us: \(2^3 \times 3^1 \times x^5 = 8 \times 3 \times x^5 = 24x^5\).
04

Write the Least Common Multiple

The least common multiple (LCM) of the two polynomials is \(24x^5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

polynomial factorization
The first step in finding the least common multiple (LCM) of polynomials is polynomial factorization. This involves breaking down each polynomial into its simpler, prime factors.
For example, consider the polynomials 8x虏 and 12x鈦�. Let's factorize them:
8x虏 can be expressed as 2鲁 * x虏.
12x鈦� can be expressed as 2虏 * 3 * x鈦�.
This way, we identify the basic building blocks of each polynomial, making the next steps easier.
highest powers
Next, we need to identify the highest powers of each factor from the factorized polynomials. This involves comparing the exponents or 'powers' of each prime factor.
For our examples, we compare the following:
For the factor 2, we have 2鲁 from 8x虏 and 2虏 from 12x鈦�. The highest power is 2鲁.
For the factor 3, there is only one instance: 3鹿 from 12x鈦�. So, the highest power is 3鹿.
For the factor x, we have x虏 from 8x虏 and x鈦� from 12x鈦�. The highest power is x鈦�.
multiplying factors
After identifying the highest powers of each factor, the next step is to multiply these highest powers together to form a new polynomial.
Using our example, we multiply the highest powers together:
2鲁 * 3鹿 * x鈦�.
To get the final result, we perform the multiplication:
2鲁 equals 8.
3鹿 equals 3.
x鈦� remains x鈦�.
So, when we multiply them: 8 * 3 * x鈦� = 24x鈦�.
least common multiple (LCM)
Finally, we arrive at the least common multiple (LCM) of the polynomials. The LCM is essentially the product obtained from Step 3, which includes the highest powers of all prime factors.
From our example, the LCM of 8x虏 and 12x鈦� is 24x鈦�.
This means that 24x鈦� is the smallest polynomial that both 8x虏 and 12x鈦� can divide without leaving a remainder.
Understanding how to find the LCM of polynomials can help simplify and solve many algebraic problems efficiently.
Remember these key concepts, and you'll find the process straightforward and intuitive!

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