Problem Analysis:
The task involves simplifying polynomial expressions and then performing operations (addition or subtraction) on the results. The specific problem highlighted in the image is:
1. Simplify the expression:
\[
(7x^2 - 60x - 27) - (5x^2 + 8x + 3)
\]
2. Solve the subtraction and simplify the result.
Step-by-Step Solution:
#### Step 1: Write down the given expression
The expression to simplify is:
\[
(7x^2 - 60x - 27) - (5x^2 + 8x + 3)
\]
#### Step 2: Distribute the negative sign
When subtracting a polynomial, distribute the negative sign to each term inside the parentheses:
\[
7x^2 - 60x - 27 - 5x^2 - 8x - 3
\]
#### Step 3: Combine like terms
Group the terms by their degrees (powers of \(x\)):
-
Quadratic terms (\(x^2\)):
\[
7x^2 - 5x^2 = 2x^2
\]
-
Linear terms (\(x\)):
\[
-60x - 8x = -68x
\]
-
Constant terms:
\[
-27 - 3 = -30
\]
#### Step 4: Write the simplified expression
Combine all the simplified terms:
\[
2x^2 - 68x - 30
\]
Final Answer:
The simplified result of the expression is:
\[
\boxed{2x^2 - 68x - 30}
\]
Explanation of the Process:
1.
Distributing the Negative Sign: This step ensures that each term inside the second polynomial is correctly negated.
2.
Combining Like Terms: Grouping terms by their powers of \(x\) helps organize the simplification process and avoids errors.
3.
Final Expression: After combining all like terms, the result is a simplified polynomial.
This method is consistent with the rules of polynomial arithmetic and ensures accuracy in solving similar problems.
Parent Tip: Review the logic above to help your child master the concept of multiply polynomials worksheet.
