Problem 30 Simplify each polynomial. See Ex... [FREE SOLUTION]

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Intermediate Algebra

Alan S. Tussy, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 5: Problem 30

Simplify each polynomial. See Example 5 . $$ 8 m^{3}-5 m^{2}-5+3 m^{2}-7 m^{3} $$

Short Answer

Expert verified

The simplified form is $ m^3 - 2m^2 - 5 $.

Step by step solution

Identify Like Terms

Like terms in a polynomial are those that have the same variable raised to the same power. In the given polynomial, identify terms with the same degree of the variable $m$: - $8m^3$ and $-7m^3$- $-5m^2$ and $+3m^2$- $-5$ is a constant term.

Combine Like Terms for Cubic Terms

Combine the cubic terms $8m^3$ and $-7m^3$: $8m^3 - 7m^3 = (8-7)m^3 = 1m^3$, which simplifies to $m^3$.

Combine Like Terms for Quadratic Terms

Combine the quadratic terms $-5m^2$ and $+3m^2$: $-5m^2 + 3m^2 = (-5+3)m^2 = -2m^2$.

Rewrite Simplified Polynomial

Now rewrite the polynomial using the simplified terms from Steps 2 and 3:\[ m^3 - 2m^2 - 5 \].

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Key Concepts

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

Like Terms

When working with polynomials, **like terms** are an essential concept. These are terms that contain the same variables raised to the same power. Identifying like terms is the first step in polynomial simplification. In our polynomial, we look for terms that can be grouped together:

Consider the terms with the variable raised to the power of three: $8m^3$ and $-7m^3$. Both terms have the variable $m$ raised to the power of three.
Look at the terms with the variable raised to the power of two: $-5m^2$ and $+3m^2$.
Finally, constants can also be considered like terms, though in this case, there is only one such term $-5$.

Recognizing like terms allows us to organize the polynomial into simpler, more manageable pieces before combining them.

Combining Like Terms

Once like terms are identified, the **combining like terms** process can begin. This step involves adding or subtracting the coefficients of the like terms.

For cubic terms, combine $8m^3$ and $-7m^3$. Calculate their sum by focusing on their coefficients: $8 - 7 = 1$. Thus, the terms simplify to $1m^3$ or simply $m^3$.
For quadratic terms, combine $-5m^2$ and $+3m^2$: $-5 + 3 = -2$. Therefore, the simplified term is $-2m^2$.

Combining like terms helps reduce the original polynomial to one with fewer terms, making it easier to handle or analyze. Each step brings you closer to the final result by streamlining complex expressions.

Simplified Polynomial

The goal of polynomial simplification is to arrive at a **simplified polynomial**. This is a cleaner version of the original polynomial where like terms have been combined. In our case, after combining the terms, we achieved:

$m^3$ from the cubic terms.
$-2m^2$ from the quadratic terms.
The constant term remains $-5$.

This results in the simplified polynomial: \[ m^3 - 2m^2 - 5 \]. Achieving a simplified polynomial is not just about reducing clutter, but also about making the polynomial easier to interpret and work with in further mathematical operations or evaluations.

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91影视

Simplify each polynomial. See Example 5 . $$ 8 m^{3}-5 m^{2}-5+3 m^{2}-7 m^{3} $$

Short Answer

Step by step solution

Identify Like Terms

Combine Like Terms for Cubic Terms

Combine Like Terms for Quadratic Terms

Rewrite Simplified Polynomial

Key Concepts

Like Terms

Combining Like Terms

Simplified Polynomial

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