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Chapter 6: Problem 26

Find the least common multiple of each pair of polynomials. $$ 15 y, 18 y^{3} $$

Short Answer

Expert verified
The least common multiple is \(90y^3\).

Step by step solution

01

Factorize Each Polynomial

First, we need to factorize each polynomial into its prime factors.The polynomial \(15y\) can be factorized into:\[15y = 3 \times 5 \times y\]The polynomial \(18y^3\) can be factorized into:\[18y^3 = 2 \times 3^2 \times y^3\]
02

Identify Highest Powers of Common and Unique Factors

To find the least common multiple (LCM), take the highest power of each prime factor that appears in the factorizations.Factors identified:\[2, 3, 5, y\]Highest powers:\[2^1, 3^2, 5^1, y^3\]
03

Multiply the Highest Powers Together

Multiply the highest powers of each factor together to find the LCM of the polynomials.\[ \text{LCM} = 2^1 \times 3^2 \times 5^1 \times y^3 \ \text{LCM} = 2 \times 9 \times 5 \times y^3 \ \text{LCM} = 90y^3 \]

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Key Concepts

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

polynomial factorization
Polynomial factorization is the process of breaking down a polynomial into simpler terms, or factors, that when multiplied together give the original polynomial. This step is crucial in finding the least common multiple (LCM) of polynomials. Factoring helps identify the building blocks of the polynomial, making it easier to work with.

For example, let's factorize the polynomials given in the exercise:
  • The polynomial \(15y\) factors into \(3 \times 5 \times y\).
  • The polynomial \(18y^3\) factors into \(2 \times 3^2 \times y^3\).
Another way to think about it is as breaking down the numbers and variables in the polynomial to their prime components and the simplest variable expressions.

Once factored, you can easily see what elements each polynomial includes, which is essential for the next steps.
highest powers of factors
For finding the LCM of polynomials, the next important step is identifying the highest powers of each factor. Once you have factored each polynomial, look at the powers of each unique variable and number that appears.

From our example:
  • For the factor \(2\), it only appears in \(18y^3\) with a power of 1 \(2^1\).
  • For the factor \(3\), it appears in both polynomials, but \(18y^3\) has the highest power with \(3^2\).
  • For the factor \(5\), it only appears in \(15y\) with a power of 1 \(5^1\).
  • For the variable \(y\), it's respective highest power is \(y^3\), as found in \(18y^3\).
By identifying the highest powers of each factor, we are ensured to cover all parts necessary for forming the LCM of these polynomials effectively.
LCM calculation
The final step is calculating the least common multiple (LCM) by multiplying the highest powers of every identified factor. This guarantees that the LCM covers all the essential elements of each polynomial, without redundant multipliers.

Here's how you do it for the example:
  • Take all the highest powers from the previous step:
  • \[2^1, 3^2, 5^1, y^3\]
  • Multiply them together: \[ \text{LCM} = 2^1 \times 3^2 \times 5^1 \times y^3 \]
Simplifying this, you get:
\[ \text{LCM} = 2 \times 9 \times 5 \times y^3 \]
Calculate it step by step:
\[ 2 \times 9 = 18 \]
\[ 18 \times 5 = 90 \]
\[ 90 \times y^3 = 90y^3 \]
Therefore, the LCM of the polynomials \(15y\) and \(18y^3\) is \(90y^3\). This means \(90y^3\) is the smallest polynomial that both \(15y\) and \(18y^3\) can divide without a remainder.

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Most popular questions from this chapter

Find the variation constant and an equation of variation in which \(y\) varies inversely as \(x,\) and the following conditions exist. \(y=27\) when \(x=\frac{1}{3}\)

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