Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation(s) and write the resulting polynomial in standard form.\(\left(15 x^{4}-18 x-19\right)-\left(13 x^{4}-5 x+15\right)\)

Step-by-Step Solution

Answer

The expression \(\left(15 x^{4}-18 x-19\right)-\left(13 x^{4}-5 x+15\right)\) simplifies to \(2x^{4} - 13x - 34\).

Step 1: Distribute the Minus Sign to the Second Polynomial

Firstly, one must distribute the minus sign to each term of the second polynomial. This will give: \(15 x^{4} - 18 x - 19 - (13 x^{4} - 5 x + 15)\) which can be simplified to \(15 x^{4} - 18 x - 19 - 13 x^{4} + 5 x - 15\). It's important to remember that the minus sign changes all the signs of the terms in the second polynomial.

Step 2: Combine Like Terms

Next, group the like terms together, i.e., those with the same power of \(x\). This will look like: \((15 x^{4} - 13 x^{4}) + (-18 x + 5 x) + (- 19 - 15)\).

Step 3: Simplify the Polynomial

By performing the calculations within the groupings, the polynomial can be simplified to give: \(2x^{4} - 13x - 34\).

Step 4: Write in Standard Form

The simplified expression is already in standard form because the terms are in decreasing order according to their degree. The standard form for a polynomial is to order the terms from highest degree to lowest.

Key Concepts

Subtraction of Polynomials

When subtracting polynomials, the key is to carefully distribute the negative sign across the terms of the polynomial being subtracted. This can be a bit tricky, so it’s important to pay close attention. Let’s break it down:

• The initial step is to write out the polynomials, ensuring they are clear and able to be operated on individually.
• For example, in the exercise, you have \((15x^4 - 18x - 19) - (13x^4 - 5x + 15)\).
• The minus operation affects each term individually, so distribute it across the second polynomial: \(13x^4 - 5x + 15 \).

Remember, subtraction here means changing each term's sign:
\(13x^4 \rightarrow -13x^4\), \(-5x \rightarrow +5x\), and \(15 \rightarrow -15\).
Repeated practice with this step will help reinforce your understanding.

Combining Like Terms

Combining like terms is an essential skill in simplifying polynomials. Each term in a polynomial is composed of a coefficient, a variable base, and an exponent. Like terms have exactly the same variable parts.

• In our exercise, after distribution, we have terms like \(15x^4\) and \(-13x^4\), which are both terms involving \(x^4\).
• Grouping these helps us focus on simplifying similar components: \((15x^4 - 13x^4)\) results in \(2x^4\).
• Similarly, group \(-18x\) and \(+5x\), alongside the constants \(-19\) and \(-15\).

This method helps simplify the expression by effectively reducing the number of terms. It is useful for visual clarity and ensuring the polynomial is as simple as possible.

Standard Form of Polynomials

The standard form of a polynomial is crucial for organizing terms correctly for comparison, addition, or further operations. This form requires that polynomials be presented with terms in decreasing order of their degree.

• After determining the like terms in the exercise and combining them, we arrived at \(2x^4 - 13x - 34\).
• This expression is in standard form because the \(x^4\) term is first, followed by the \(x\) term, and finally the constant term, with no terms of the same degree left unsimplified.

The organization helps quickly identify the leading term and degree of the polynomial, which is immensely useful in longer algebraic calculations or when you need to graph polynomial functions on a coordinate plane.