Problem 51 \(\left(x^{3}-x^{2}-4\right)-\le... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

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Chapter 5: Problem 51

$\left(x^{3}-x^{2}-4\right)-\left(x^{3}-x-4\right)$

Short Answer

Expert verified

-x^{2} + x

Step by step solution

- Understand the problem

The given expression is $\backslash ( x^{3}-x^{2}-4 \backslash ) $- $ \backslash ( x^{3}-x-4 \backslash ) $. To solve, we need to simplify this by removing the parentheses and combining like terms.

- Distribute the negative sign

Distribute the negative sign across the second set of parentheses: $ x^{3}-x^{2}-4 - (x^{3}-x-4) \Rightarrow x^{3}-x^{2}-4 - x^{3} + x + 4 $.

- Combine like terms

Combine the terms with the same power of $ x $: $ x^{3} - x^{3} - x^{2} + x - 4 + 4 $. This simplifies to: \ -x^{2} + x \.

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Key Concepts

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

Polynomial Subtraction

Polynomial subtraction may sound complicated, but it's straightforward once understood. In our exercise, we began with the given expression: \ $ \left( x^{3}-x^{2}-4 \right) - \left( x^{3}-x-4 \right) $.
When subtracting polynomials, you must pay close attention to each term in both expressions, and, importantly, correctly distribute the negative sign from the subtraction. This ensures you accurately identify and combine terms with the same power.

It's like dealing with two groups of terms where you need to carefully subtract each term in the second polynomial from the corresponding term in the first.

Distributing the Negative Sign

When you subtract polynomials, distributing the negative sign is crucial. This step flips the signs of all the terms inside the second set of parentheses.

In our example, we started with:
\[ x^{3}-x^{2}-4 - \left( x^{3}-x-4 \right) \]

Distributing the negative sign we get:
\[ x^{3}-x^{2}-4 - x^{3} + x + 4 \]
Notice how every term in the second polynomial has changed its sign.

Combining Like Terms

The final step to simplifying polynomial expressions is combining like terms. Like terms have the same power of $ x $.

In our example:
\[ x^{3} - x^{3} - x^{2} + x - 4 + 4 \]

We combine them as follows:

$ x^{3} - x^{3} = 0 $
$ - x^{2} $ remains as it doesn't have a counterpart
$ + x $ remains for the same reason
$ - 4 + 4 = 0 $

The simplified expression is: \[ -x^{2} + x \]
Combining like terms efficiently cuts down on complexity and zeroes in on the essential part of the polynomial.

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91影视

\(\left(x^{3}-x^{2}-4\right)-\left(x^{3}-x-4\right)\)

Short Answer

Step by step solution

- Understand the problem

- Distribute the negative sign

- Combine like terms

Key Concepts

Polynomial Subtraction

Distributing the Negative Sign

Combining Like Terms

One App. One Place for Learning.

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