Problem 38 Factor each monomial. $$-120 r... [FREE SOLUTION]

Chapter 4: Problem 38

Factor each monomial. $$-120 r^{2} s t^{3}$$

Short Answer

Expert verified

The monomial is factored as $-1 \times 2^3 \times 3 \times 5 \times r^2 \times s \times t^3$.

Step by step solution

Identify Numerical Factors

Break down the numerical part of the monomial. The number $-120$ can be factored into prime numbers: $-120 = -1 \times 2^3 \times 3 \times 5$. The factor $-1$ is included as it's important to represent the sign of the number.

Identify Variable Factors

List out the variable parts of the monomial along with their exponents: $r^{2}, s, t^{3}$. Each variable is a factor, with their exponents indicating how many times they appear as a factor.

Combine All Factors

Combine all the factors from the numerical and variable parts of the monomial together: $-1 \times 2^3 \times 3 \times 5 \times r^2 \times s \times t^3$.

Write the Factorization

Write the factorization succinctly by including all identified factors. The completely factored form is: $-1 \times 2^3 \times 3 \times 5 \times r^2 \times s \times t^3$.

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Key Concepts

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

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. This is a key concept in finding factors of more complex numbers, such as the numerical part of a monomial. For example, let's break down $-120$. The prime factorization is achieved by dividing the number into prime factors:

Start by dividing $-120$ by the smallest prime number, which is 2. Divide repeatedly until the number can no longer be divided by 2. Here, $-120 = -1 \times 2^3$.

Next, look for the next smallest prime number. For $-120$, it's 3. So we have $-1 \times 2^3 \times 3$.

Finally, check the remaining number, which in this step is 5. Since 5 is a prime number itself, it concludes our factorization as $-1 \times 2^3 \times 3 \times 5$.

The negative sign can be represented by the factor $-1$. Each prime serves as a building block of the number, much like atoms in a molecule.

Exponents in Monomials

Exponents in monomials indicate how many times a factor is repeated. They compactly express repeated multiplication, making algebraic evaluations simpler and more efficient. In our monomial $-120 r^2 s t^3$, each variable is powered by an exponent:

$r^2$ tells us that the variable $r$ is used twice as a factor.

$s$ appears without an exponent, meaning it is used once.

$t^3$ implies $t$ is a factor three times.

When working with exponents, it is vital to remember that exponents apply only to the immediate variable or number unless parentheses indicate otherwise. For example, in $ (ab)^2$, the exponent 2 applies to both $a$ and $b$. This approach simplifies the expression of large numbers or variables repeated in a factorization.

Algebraic Expressions

Algebraic expressions use numbers, variables, and arithmetic operations to express a mathematical phrase. Expressions can be expanded, factored, or simplified to serve specific mathematical purposes. A monomial is a specific type of algebraic expression containing only one term. It's a product of numbers and variables, each raised to a power, like our exercise's example: $-120 r^2 s t^3$.

Algebraic expressions are as versatile as they are foundational because they can represent various real-world phenomena, such as physical constants or economic formulas.

Monomials simplify the representation of expressions involving powers and multiple factors, making them easier to manipulate in mathematical operations.

Understanding the structure of algebraic expressions aids in operations like adding, subtracting, and factoring complex mathematical equations, centralizing the role of variables and constants in calculations.

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91影视

Factor each monomial. $$-120 r^{2} s t^{3}$$

Short Answer

Step by step solution

Identify Numerical Factors

Identify Variable Factors

Combine All Factors

Write the Factorization

Key Concepts

Prime Factorization

Exponents in Monomials

Algebraic Expressions

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