Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation(s) and write the resulting polynomial in standard form.\(-\left(5 x^{2}-1\right)+\left(-3 x^{2}+5\right)\)

Step-by-Step Solution

Answer

\(-8x^{2} + 6\)

Step 1: Apply The Negative Sign To The First Polynomial

Applying the negative sign to every term in the first bracket we get:\(-5x^{2} + 1\). Hence, the expression now looks like: \(-5x^{2} + 1 -3x^{2} + 5\)

Step 2: Combine Like Terms

Next, we need to add the terms with the same variable and power. In this expression that would be \(-5x^{2}\) and \(-3x^{2}\) as well as the constant terms \(1\) and \(5\). These additions we get: \(-5x^{2} - 3x^{2} + 1 + 5\)

Step 3: Perform The Additions

Finally, add the terms from the previous step to get the resulting polynomial. The mathematic formular applied here is \(-5 -3 = -8\) and \(1 + 5 = 6\). The resulting polynomial in standard form is: \(-8x^{2} + 6\)

Key Concepts

Standard Form

When working with polynomials, writing them in standard form is essential for ease of reading and further calculations. The standard form of a polynomial arranges terms in descending order of their degree. This means starting with the highest power of the variable and working down to the constant. For instance, a polynomial like \(ax^n + bx^{n-1} + \,\ldots\,+ zx^0\) is in standard form because each term appears in order of descending powers of the variable \(x\).
Writing polynomials in standard form provides a consistent format that simplifies operations such as addition, subtraction, and solving equations. It helps in identifying like terms, analyzing the polynomial's behavior, and comparing different polynomials effectively.
For example, the expression \(-8x^2 + 6\) is in standard form because the term \(-8x^2\) has the highest degree (2) and is followed by the constant \(6\). This order is crucial to ensure clarity in mathematical procedures.

Combining Like Terms

The concept of combining like terms is crucial in simplifying polynomials. It involves merging terms that have the exact same variable raised to the same power. Consider the expression from the exercise: \(-5x^2 - 3x^2 + 1 + 5\). Here, \(-5x^2\) and \(-3x^2\) are like terms because they both contain \(x^2\).
When combining like terms:

• Add or subtract the coefficients of these terms, keeping the variable and its exponent unchanged.
• Combine constant terms (numbers without variables) separately.

In the given exercise, combining \(-5x^2\) and \(-3x^2\) yields \(-8x^2\). Similarly, combining the constants \(1\) and \(5\) gives \(6\). The resulting polynomial is \(-8x^2 + 6\). This process is vital as it reduces expressions to their simplest and most manageable form.

Negative Sign Distribution

Applying a negative sign correctly is key when simplifying polynomial expressions, especially when dealing with subtraction. Negative sign distribution involves applying the negative sign to each term within parentheses or a set of brackets.
In our exercise, the expression initially has \(-\left(5x^2 - 1\right)\). Distribute the negative sign to change the signs of the enclosed terms. This transforms it into \(-5x^2 + 1\). The correct application of this process ensures that all terms are accurate for further operations like combining like terms.
Steps to apply negative sign distribution:

• Each positive term turns negative.
• Each negative term turns positive.

Understanding this concept avoids errors in calculation and ensures the manipulation of polynomials is done efficiently and correctly. Make sure to double-check this step before moving on to combining terms.