Problem 491 Divide each polynomial by the bi... [FREE SOLUTION]

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Chapter 6: Problem 491

Divide each polynomial by the binomial. $$\left(m^{3}+1000\right) \div(m+10)$$

Short Answer

Expert verified

$ m^2 - 10m + 100 $

Step by step solution

Identify the Polynomial and Binomial

Given polynomial is $ m^3 + 1000 $ and binomial is $ m + 10 $.

Express the Polynomial in Factorized Form

Notice that $ m^3 + 1000 $ is a sum of cubes. Use the sum of cubes formula: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] Here, $ a = m $ and $ b = 10 $. Therefore: \[ m^3 + 1000 = (m + 10)(m^2 - 10m + 100) \]

Perform the Division

Since the polynomial $ m^3 + 1000 $ is $ (m + 10)(m^2 - 10m + 100) $, dividing it by $ m + 10 $ cancels the $ m + 10 $ terms: \[ \frac{(m + 10)(m^2 - 10m + 100)}{m + 10} = m^2 - 10m + 100 \]

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Key Concepts

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

Polynomial Division

When dividing polynomials, the process is similar to long division with numbers. The goal is to simplify the given polynomial by breaking it into smaller, more manageable parts. For example, consider the polynomial given in the exercise: $$ m^3 + 1000. The problem requires us to divide this by the binomial $$ m + 10. A helpful strategy in polynomial division is understanding and using factorization.

Binomials

A binomial is a polynomial with exactly two terms. In our exercise, $$ m + 10 is the binomial. Recognizing binomials is important since they often serve as the simplest building blocks in polynomial division. When dividing a polynomial by a binomial, you look for common factors that can simplify the process. For instance, if you identify that your polynomial can be expressed as a product of the binomial and another polynomial, the division becomes straightforward.

Sum of Cubes

The sum of cubes is a specific polynomial form where two cubed terms are added together. In mathematical terms, this can be represented as $$ a^3 + b^3. In our problem, $$ m^3 + 1000 is a sum of cubes where $$ a = m and $$ b = 10. The sum of cubes can be factorized using the following formula: $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2). For our exercise, applying this formula shows us that: $$ m^3 + 1000 = (m + 10)(m^2 - 10m + 100). After factorizing the polynomial, dividing it by the binomial $$ m + 10 becomes simple as the $$ m + 10 terms cancel out, leaving us with $$ m^2 - 10m + 100.

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91影视

Divide each polynomial by the binomial. $$\left(m^{3}+1000\right) \div(m+10)$$

Short Answer

Step by step solution

Identify the Polynomial and Binomial

Express the Polynomial in Factorized Form

Perform the Division

Key Concepts

Polynomial Division

Binomials

Sum of Cubes

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