Problem 38 - College Algebra with Modeling and Visualization

Question

Subtract the polynomials. $$\left(-x^{2}+x-5\right)-\left(x^{2}-x+5\right)$$

Step-by-Step Solution

Answer

The result is \(-2x^2 + 2x - 10\).

Step 1: Distribute the Negative Sign

To subtract the second polynomial \(x^2 - x + 5\) from the first polynomial, distribute the negative sign across each term in the second polynomial. This changes the subtraction into addition with the opposite signs.\[(-x^2 + x - 5) - (x^2 - x + 5) = -x^2 + x - 5 - x^2 + x - 5\]

Step 2: Combine Like Terms

Next, combine like terms. Group the similar terms together: terms with \(x^2\), terms with \(x\), and constant terms.1. For \(x^2\) terms: \(-x^2 - x^2 = -2x^2\)2. For \(x\) terms: \(x + x = 2x\)3. For constant terms: \(-5 - 5 = -10\)Combine these results.\[-2x^2 + 2x - 10\]

Step 3: Write the Simplified Polynomial

Write down the simplified result after combining all like terms from the previous step.\(-2x^2 + 2x - 10\)

Key Concepts

Distribute the Negative Sign

When subtracting one polynomial from another, the process is a bit more than merely changing the sign in front of each term. This requires distributing a negative sign to every term inside the parentheses of the second polynomial. Consider the polynomial subtraction problem: \[(-x^2 + x - 5) - (x^2 - x + 5)\]
To proceed, change the signs of each term in the second polynomial:

• The first term, \(x^2\), becomes \(-x^2\).
• The second term, \(-x\), changes to \(+x\).
• The third term, \(+5\), becomes \(-5\).

Once the signs are switched, the expression becomes:\[-x^2 + x - 5 - x^2 + x - 5\]This step is vital because it ensures that each component of the second polynomial is correctly subtracted from the first.

Combine Like Terms

After altering the signs and thus adjusting the polynomial expression, the next task is to combine like terms. Like terms are any terms that include the same power of the variable. Here's how it goes with the expression obtained from the previous step:

• Combine the \(x^2\) terms: \(-x^2 - x^2 = -2x^2\).
• Combine the \(x\) terms: \(x + x = 2x\).
• Combine the constant terms: \(-5 - 5 = -10\).

This simplifies the expression significantly, leading us to:\[-2x^2 + 2x - 10\]
By grouping and adding or subtracting like terms, we ensure the polynomial is as simplified as possible before the final step.

Simplify Polynomial Expressions

The final goal in subtracting and simplifying polynomials is to ensure that the expression is reduced to its simplest form. This means all like terms have been combined, and the polynomial is written neatly. From the previous steps, we reached:\[-2x^2 + 2x - 10\]
This expression is already simplified because:

• It accurately reflects the subtraction of the original polynomials.
• All terms have been simplified, and no further reduction is possible.
• There is clear organization with terms lined by their degree, from highest to lowest.

The journey to simplifying polynomials involves working through each step methodically to arrive at clarity and precision in your expressions.