Multiplying Polynomials
Welcome to today's lesson on multiplying polynomials! We'll explore how to multiply monomials, binomials, and trinomials using the distributive property. Get ready to expand your algebraic toolkit!
Key Concepts
- Monomial: A single-term expression like $2ab^2$.
- Binomial: A two-term expression like $(x + 2)$.
- Trinomial: A three-term expression like $(4x^2 - 3x - 1)$.
Multiplying a Monomial and a Polynomial
The distributive property is key when multiplying a monomial by a polynomial. For example:
$$2ab^2(3a^2b - 4ab^2 - b^3)$$
Distribute $2ab^2$ to each term inside the parentheses:
$$2ab^2(3a^2b) + 2ab^2(-4ab^2) + 2ab^2(-b^3)$$ $$= 6a^3b^3 - 8a^2b^4 - 2ab^5$$
Remember to add the exponents when multiplying like bases!
Multiplying Two Polynomials
When multiplying two polynomials (e.g., two binomials or a binomial and a trinomial), distribute each term of the first polynomial to each term of the second polynomial. There are two common methods:
1. Horizontal Method:
Example: $(x + 2)(4x^2 - 3x - 1)$
Distribute each term:
$$x(4x^2 - 3x - 1) + 2(4x^2 - 3x - 1)$$ $$= (4x^3 - 3x^2 - x) + (8x^2 - 6x - 2)$$ $$= 4x^3 + 5x^2 - 7x - 2$$
2. Vertical Method:
Align like terms to simplify the addition process. This is similar to long multiplication:
4x² - 3x - 1
x + 2
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8x² - 6x - 2 (Multiply by 2)
4x³ - 3x² - x (Multiply by x)
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4x³ + 5x² - 7x - 2 (Combine like terms)
Practice Problems
Try these practice problems to reinforce your understanding:
- $4x^2(x^2 + 2x - 3)$
- $(x - 3)(x^2 - 2x + 2)$
Class Assignments
- Homework: Pg. 162 # 1-8
Keep practicing, and you'll master polynomial multiplication in no time! Remember to take your time and double-check your work. Good luck!