Problem 41 - College Algebra with Modeling and Visualization

Question

Subtract the polynomials. $$\left(x^{4}-1\right)-\left(4 x^{4}+3 x+7\right)$$

Step-by-Step Solution

Answer

The simplified polynomial is \(-3x^4 - 3x - 8\).

Step 1: Rewrite the Problem

First, rewrite the subtraction of polynomials as a sum, distributing the minus sign to each term in the second polynomial. The expression becomes: \[(x^4 - 1) - (4x^4 + 3x + 7) = x^4 - 1 - 4x^4 - 3x - 7\]

Step 2: Combine Like Terms

Now, let's combine the like terms by aligning them: - The terms with \(x^4\) are \(x^4 - 4x^4\).- The terms with \(x\) appear only once as \(-3x\).- The constant terms are \(-1 - 7\).Simplifying each group results in:\((x^4 - 4x^4) + (-3x) + (-1 - 7)\).

Step 3: Simplify the Expression

Finally, simplify each group of like terms:- For \(x^4\), we have \(x^4 - 4x^4 = -3x^4\).- The \(x\) term remains \(-3x\).- The constant terms are \(-1 - 7 = -8\).Put these together to get the simplified expression: \(-3x^4 - 3x - 8\).

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables, coefficients, and exponents. They can be as simple as a single term like \(x\), or consist of many terms combined with operations like addition, subtraction, and multiplication. Each term in a polynomial is made up of:

• Variables: Symbols like \(x\) that can represent different numbers.
• Coefficients: Numbers that multiply the variables.
• Exponents: Numbers that indicate how many times the variable is multiplied by itself.

A polynomial can have different degrees, based on the highest exponent of the variable present. The example given in the exercise involves two polynomials, which means we are engaging in the process of manipulating more than one polynomial to reach a solution. By understanding how polynomials are structured, we can easily manage operations like addition, subtraction, and more.

Like Terms

Like terms are an essential concept when dealing with polynomials, as they are terms that contain the same variable raised to the same power. Only like terms can be combined through addition or subtraction. This is a key step in simplifying expressions. When you look at the expression from the exercise, \((x^4 - 1) - (4x^4 + 3x + 7)\), you'll notice terms such as \(x^4\) and \(4x^4\), which are considered like terms because they share the same variable and exponent. Similarly, \(-3x\) is a separate term with a different exponent, and the constants \(-1\) and \(-7\) are also like terms because they have the same effective power of zero. To manage like terms efficiently, remember:

• Align terms with the same variables and powers.
• Combine them through basic arithmetic operations.

Recognizing and combining like terms leads to a simplified and very readable polynomial expression.

Simplification

Simplification is the process of reducing a polynomial expression to its simplest form. This involves combining like terms and applying basic arithmetic to express the polynomial more compactly. In the exercise, we began with the expression: \[x^4 - 1 - 4x^4 - 3x - 7\], which on simplification leads to a more straightforward result.Here are some strategies for effective simplification:

• Distribute negative signs or coefficients through the expression. This ensures all terms have the correct sign when subtracted or added, as seen when the \(-\) sign was distributed in the step where \((4x^4 + 3x + 7)\) became \(-4x^4 - 3x - 7\).
• Carefully combine like terms. For example, \(x^4 - 4x^4\) simplifies to \(-3x^4\).
• Perform arithmetic on constant terms. Combine numbers like \(-1\) and \(-7\) to simplify to \(-8\).

The final expression \(-3x^4 - 3x - 8\) represents the simplest form, making it much easier to work with or further analyze. Simplification not only shortens expressions but also reveals the true nature of the polynomial through reduced and clear terms.