BackAdding and Subtracting Polynomials: Integrated Review
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Polynomial Functions and Operations
Adding and Subtracting Polynomials
Polynomials are algebraic expressions consisting of terms with variables raised to whole number exponents. Understanding how to add and subtract polynomials is fundamental in College Algebra, as it forms the basis for more advanced operations and problem-solving.
Like Terms: Terms with the same variable(s) raised to the same power(s) are called like terms. Only like terms can be combined through addition or subtraction.
Combining Like Terms: To add or subtract polynomials, collect like terms and perform the indicated operation on their coefficients.
Adding Polynomials
To add polynomials, write them together with a plus sign and combine like terms using the distributive property.
Step 1: Align like terms from each polynomial.
Step 2: Add the coefficients of like terms.
Step 3: Write the simplified expression.
Example 1: Add
Collect like terms:
Simplify:
Example 2: Add
Collect like terms:
Simplify:
Additional info: The notation "f^*" and "r" may refer to variables; clarify based on context.
Subtracting Polynomials
To subtract a polynomial, add its opposite. The opposite of a polynomial is found by changing the sign of each term.
Step 1: Change the sign of each term in the polynomial being subtracted.
Step 2: Add the resulting polynomial to the original.
Step 3: Combine like terms and simplify.
Example 3: Subtract
Add the opposite:
Collect like terms:
Simplify:
Example 4: Subtract
Add the opposite:
Collect like terms:
Simplify:
Example 5: Subtract
Add the opposite:
Collect like terms:
Simplify:
Key Properties
Distributive Law:
Commutative Law:
Associative Law:
Summary Table: Steps for Adding and Subtracting Polynomials
Operation | Step 1 | Step 2 | Step 3 |
|---|---|---|---|
Addition | Align like terms | Add coefficients | Simplify expression |
Subtraction | Find opposite of second polynomial | Add to first polynomial | Combine like terms and simplify |
Additional info: The examples provided demonstrate the process of combining like terms and using the distributive property to simplify polynomial expressions. These skills are foundational for further study in polynomial functions and algebraic manipulation.
