Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation(s) and write the resulting polynomial in standard form.\(-\left(x^{3}+5\right)+\left(3 x^{3}-4 x\right)\)

Step-by-Step Solution

Answer

The resulting polynomial in standard form is \(2x^{3} - 4x - 5\).

Step 1: Distribute the negative

The first thing to do is the distribution of the negative sign on the first expression. This changes the signs of all terms within the parentheses. So, \(-\left(x^{3}+5\right)\) becomes \(-x^{3}-5\).

Step 2: Combine like terms

Next, combine like terms from each expression. Therefore, combine \(-x^{3}\) from the first expression with \(3x^{3}\) from the second expression. Also, take note of the \(-5\) and \(-4x\) remaining, as they do not have any like terms they could be combined with. This gives \(3x^{3} - x^{3} - 4x - 5\).  

Step 3: Simplify the expression

Simplify the polynomial resulting from the combination of like terms. \(3x^{3} - x^{3} = 2x^{3}\). Hence, the polynomial in standard form is \(2x^{3} - 4x - 5\).

Key Concepts

Standard Form

A polynomial is in standard form when its terms are written in descending order of their degrees, starting with the term that has the highest exponent. This is important because it makes the polynomial easy to read and work with, especially when comparing or combining them.

For instance, when looking at a polynomial like \(2x^3 - 4x - 5\), it is in standard form because the term with the highest power, \(2x^3\), comes first. Each subsequent term has a smaller degree, and the constant term is last.

Putting polynomials in standard form helps in identifying like terms and supports neat and clear computations during operations like addition, subtraction, and multiplication.

Combining Like Terms

Combining like terms involves summing the coefficients of terms that have the same variable and the same exponent. This is a crucial step that simplifies polynomials, making them easier to manage.

In the case of our example \(-x^3 + 3x^3 - 4x - 5\), the like terms are \(-x^3\) and \(3x^3\). You combine them by adding their coefficients:

• Coefficient of \(-x^3\) is -1
• Coefficient of \(3x^3\) is 3

Thus, \(3x^3 - x^3\) becomes \(2x^3\).

The remaining terms, \(-4x\) and \(-5\), have no like terms to combine with and remain unchanged. This simplification is how you end up with the simpler expression \(2x^3 - 4x - 5\).

Distributive Property

The distributive property is a vital concept in algebra used to simplify expressions and solve equations. It allows you to distribute a multiplication over addition or subtraction inside parentheses.

In the original expression \(-\left(x^{3}+5\right) + \left(3 x^{3} - 4 x\right)\), applying the distributive property involves the negative sign outside the first set of parentheses.

This means you multiply -1 by each term inside:

• \(-1 × x^3\) gives \(-x^3\)
• \(-1 × 5\) gives \(-5\)

So, \(-\left(x^{3}+5\right)\) transforms into \(-x^3 - 5\).

This step is crucial before combining like terms, ensuring the entire expression is correctly expanded and simplified.