Problem 47 - Beginning and Intermediate Algebra

Question

Subtract. $$\left(j^{2}+18 j+2\right)-\left(-7 j^{2}+6 j+2\right)$$

Step-by-Step Solution

Answer

The short answer to the given subtraction of polynomials is: \( 8j^2 + 12j \).

Step 1: Identify the corresponding terms

First, we should identify the corresponding terms in both polynomials. Corresponding terms are the terms with the same power of the variable (in this case, j). For example, the terms with j^2 are corresponding terms.

Step 2: Subtract the corresponding terms

Now that we have identified the corresponding terms in both polynomials, we need to subtract them. Here's the subtraction: \( (\,\, j^2 - (-7j^2)\,\,) + (\,\, 18j - 6j\,\,) + (\,\, 2 - 2\,\,) \)

Step 3: Perform the arithmetic operations

Now that we have set up the subtraction of corresponding terms, we can perform the arithmetic operations for each term: \( (8j^2) + (12j) + (0) \)

Step 4: Simplify the final result

Our final step is to simplify the result. Since there are no further terms to combine or simplify, the final answer is: \( 8j^2 + 12j \)

Key Concepts

Corresponding Terms

In polynomial subtraction, one of the first things we need to understand is identifying corresponding terms. Corresponding terms are terms that have the same power of the variable. These terms match each other in terms of their degree and need to be aligned before performing subtraction. In the exercise given, the polynomials are:

• \(j^{2}+18j+2\)
• \(-7j^{2}+6j+2\)

The corresponding terms in each polynomial are:

• \(j^2\) and \(-7j^2\) in both polynomials.
• \(18j\) and \(6j\).
• The constant \(2\) in both polynomials.

Recognizing these corresponding terms is essential because it sets the stage for correct subtraction across each term, ensuring no mismatch in terms.

Simplifying Polynomials

Simplifying polynomials involves combining like terms and reducing expressions to their simplest form. Once we have identified the corresponding terms, the next step is subtraction. We need to subtract each corresponding term:

• Subtract \(-7j^2\) from \(j^2\), which yields \((j^2 - (-7j^2)) = 8j^2\).
• Subtract \(6j\) from \(18j\), resulting in \((18j - 6j) = 12j\).
• Subtract the constants \(2\) from \(2\), which equals zero.

After performing these subtractions, you are left with the simplified polynomial: \(8j^2 + 12j\). Simplifying makes polynomials more manageable, providing the simplest possible form for further computations or understanding.

Arithmetic Operations with Polynomials

Arithmetic operations with polynomials include adding, subtracting, multiplying, and dividing polynomials. The subtraction of polynomials, as demonstrated in the exercise, follows a straightforward process. You align the corresponding terms and then subtract them while treating each polynomial as a whole expression:

• Understand that subtraction involves adding the negative of the second polynomial. Remember to distribute the minus sign throughout the whole polynomial if you're doing this step manually.
• Apply the basic arithmetic operation to each pair of corresponding terms: adding the negative term when subtracting.
• Combine all like terms resulting from these operations.

These arithmetic operations allow us to handle complex polynomial expressions efficiently and form the base of more advanced algebraic manipulations.