Multiplying Polynomials
Learn how to multiply polynomials using the distributive property and FOIL method.
Definition
- First terms:
- Outer terms:
- Inner terms:
- Last terms:
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Worked Examples
Multiply:
Distribute the monomial to each term
→ Apply distributive property
Multiply the first product
→
Multiply the second product
→
Combine the terms
→ Final answer
Answer:
Common Mistakes
Only multiplying the first and last terms:
Why it's wrong: Every term in the first polynomial must multiply every term in the second polynomial.
Correct: Use FOIL:
Forgetting the middle term when squaring:
Why it's wrong: Squaring a binomial creates three terms, not two. The middle term is .
Correct:
Sign errors with negative terms:
Why it's wrong: A negative times a positive is negative.
Correct:
Adding exponents incorrectly:
Why it's wrong: When multiplying variables, we add exponents:
Correct:
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Practice Problems
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Why It Matters
- Factoring: To factor, you need to understand how factors multiply
- Solving equations: Quadratic equations often come from polynomial multiplication
- Area calculations: Finding the area of rectangles with variable dimensions
- Real-world modeling: Physics, engineering, and economics use polynomial products
Real World Applications
Area of a Garden
Finding areas with variable dimensions requires polynomial multiplication.
Example:
A rectangular garden has length meters and width meters. The area is square meters.
A frame around a picture adds 2 cm to each side. The picture is cm by cm.
What is the total area including the frame?
Step 1: Write the mathematical expression
Multiply :
Business Revenue
Revenue calculations often involve multiplying expressions for price and quantity.
Example:
If price is dollars and quantity sold is items, revenue is dollars.
A shop sells items at euros each.
Write the revenue as a polynomial.
Step 1: Write the mathematical expression
Multiply :
Key Takeaways
- 1Use the distributive property to multiply each term in one polynomial by every term in the other
- 2For two binomials, use FOIL: First, Outer, Inner, Last
- 3Always combine like terms after multiplying
- 4When squaring a binomial:
- 5Watch for sign errors especially with negative terms
Frequently Asked Questions
Glossary
- Polynomial
- An expression with one or more terms involving variables and coefficients (e.g., )
- Binomial
- A polynomial with exactly two terms (e.g., )
- Trinomial
- A polynomial with exactly three terms (e.g., )
- FOIL
- A method for multiplying two binomials: First, Outer, Inner, Last
- Like terms
- Terms with the same variable raised to the same power (e.g., and )
Formula Card
Distributive Property
Multiply the term outside by each term inside
FOIL Method
First, Outer, Inner, Last for two binomials
Square of a Sum
Perfect square trinomial pattern
Square of a Difference
Note the minus sign on middle term
Difference of Squares
Middle terms cancel out