Problem 56 - Elementary Algebra

Question

\(\left(10 c^{2}-\frac{9}{10} c+8\right)-\left(7 c^{2}-\frac{1}{10} c+2\right)\)

Step-by-Step Solution

Answer

3c² - 0.8c + 6

Step 1: Distribute the negative sign

Distribute the negative sign to each term inside the second parenthesis, changing the signs of each term.\[(10c^2 - \frac{9}{10}c + 8) - (7c^2 - \frac{1}{10}c + 2)\rightarrow (10c^2 - \frac{9}{10}c + 8) - 7c^2 + \frac{1}{10}c - 2\]

Step 2: Combine like terms for the quadratic terms

Combine the quadratic terms \(10c^2\) and \(-7c^2\).\[10c^2 - 7c^2 = 3c^2\]

Step 3: Combine like terms for the linear terms

Combine the linear terms \(-\frac{9}{10}c\) and \(\frac{1}{10}c\).\[-\frac{9}{10}c + \frac{1}{10}c = -\frac{8}{10}c = -0.8c\]

Step 4: Combine like terms for the constant terms

Combine the constant terms 8 and -2.\[8 - 2 = 6\]

Step 5: Write the simplified expression

Put all the combined terms together for the final expression.\[3c^2 - 0.8c + 6\]

Key Concepts

Distribute Negative Sign

Let's tackle polynomial subtraction by first distributing the negative sign. This is a crucial step because not distributing the negative sign correctly can lead to errors. The expression to work with is: \( (10c^2 - \frac{9}{10}c + 8) - (7c^2 - \frac{1}{10}c + 2) \).
Here, we need to distribute the negative sign to each term inside the parenthesis after the subtraction sign.
This process changes the signs of each term inside the parenthesis:
\( (10c^2 - \frac{9}{10}c + 8) - 7c^2 + \frac{1}{10}c - 2 \).
This step sets up the equation for the next processes of combining like terms and simplifying. Remember, distributing the negative sign correctly ensures the overall correctness of your final answer.

Combine Like Terms

After distributing the negative sign, you need to combine like terms. Like terms are terms that have the same variables raised to the same power.
Grouping Like Terms:
We group the quadratic terms, linear terms, and constant terms separately.
For the given example: \( (10c^2 - \frac{9}{10}c + 8) - 7c^2 + \frac{1}{10}c - 2 \):

• Quadratic terms: \( 10c^2 \) and \( -7c^2 \)
• Linear terms: \( -\frac{9}{10}c \) and \( \frac{1}{10}c \)
• Constant terms: \( 8 \) and \( -2 \)

Combining Like Terms:

• For quadratic terms: \( 10c^2 - 7c^2 = 3c^2 \)
• For linear terms: \( -\frac{9}{10}c + \frac{1}{10}c = -\frac{8}{10}c = -0.8c \)
• For constant terms: \( 8 - 2 = 6 \)

Combining like terms simplifies the expression to be more manageable for the final step.

Simplify Polynomials

Once you have combined all like terms, the final step is to write the simplified expression. Organize the combined terms in standard form, which means arranging the terms from highest degree to lowest degree:
For the expressions:

• Quadratic term: \( 3c^2 \)
• Linear term: \( -0.8c \)
• Constant term: \( +6 \)

Now, combine these simplified terms:
\[ 3c^2 - 0.8c + 6 \]
And there you have it! The simplified polynomial expression.
This clean and concise method of subtraction, combining like terms, and simplification is crucial for understanding more complex algebraic processes. Keep practicing, and it will soon become second nature.