Unit 6: Quadratic and Polynomial Functions
Topic 5
Adding, Subtracting, and Multiplying Polynomials
Polynomials are algebraic expressions made up of terms that include variables raised to whole-number powers.
Examples: , , and .
Operations on polynomials follow the same rules as arithmetic, except that we combine like terms—those with the same variable and exponent.
1. Adding and Subtracting Polynomials
To add or subtract, align like terms and combine their coefficients.
2. Multiplying Polynomials
Use the Distributive Property or FOIL for binomials:
When multiplying larger polynomials, multiply each term in one by each term in the other and then combine like terms.
3. Special Products to Remember
Recognizing these patterns can save time on the SAT.
Core Skills
- Identify and combine like terms correctly.
- Apply distributive property to multiply expressions.
- Recognize and expand special product patterns.
- Simplify expressions by collecting terms and ordering by degree.
- Apply these operations in context problems (area, expansion, algebraic structure).
Example 1: Adding Polynomials
Simplify:
Combine like terms:
Result:
Example 2: Subtracting Polynomials
Simplify:
Distribute the negative:
Result:
Example 3: Multiplying Binomials (FOIL)
Simplify:
Use FOIL (First, Outer, Inner, Last):
Result:
Example 4: Difference of Squares
Simplify:
Result:
Example 5: Squaring a Binomial
Simplify:
Result:
Example 6: Area Application
The length of a rectangle is and the width is .
Find the area as a polynomial.
Area:
Key Takeaways
- Combine only like terms when adding or subtracting.
- Apply distributive property for multiplication.
- Recognize shortcuts for special products to save time.
- Always simplify and arrange terms in descending powers.
- These operations appear in both algebraic and geometric SAT contexts.