(a+/-b...)(c+/-d+/-e...) - Algebra-1

1. Fundamental Concepts

is the core form of "multiplying a polynomial by a polynomial" in polynomial multiplication. Its essence lies in the two-time application of the Distributive Property of Multiplication: First, treat each term (such as a, ) in the first polynomial as a "single term" and multiply it by all terms in the second polynomial . Then, add up all the intermediate products and finally combine like terms to get the result. Both polynomials contain 2 or more terms (they can be binomials, trinomials, etc.), and each term can be a constant or a monomial.

"Term-by-Term Multiplication" Principle: Every term in the first polynomial must be multiplied by every term in the second polynomial, with no pair of terms omitted.
Sign Rule: When multiplying each pair of terms, follow the principle of "same sign gives positive, different signs give negative". Pay attention to the original signs of both terms being multiplied. For instance, in , and .
Combining Like Terms: After multiplying all terms, identify like terms (terms with exactly the same letters and exponents, e.g., and ), combine them by adding or subtracting their coefficients, and keep non-like terms in the result.
Degree and Coefficient: When multiplying, multiply the coefficients (considering signs), add the exponents of the same letter (e.g., ), and retain different letters directly in the result.

Calculate
Solution:
Multiply term by term (cover all pairs of terms): ;
Calculate intermediate terms: ;
Combine like terms: .

Problem: $$(2y^2 + y - 1)(y + 3)$$

Distribute each term of the first polynomial to each term of the second polynomial: $$2y^2 \cdot y + 2y^2 \cdot 3 + y \cdot y + y \cdot 3 - 1 \cdot y - 1 \cdot 3$$
Simplify: $$2y^3 + 6y^2 + y^2 + 3y - y - 3$$
Combine like terms: $$2y^3 + 7y^2 + 2y - 3$$

Visual Strategy: Use a grid or box method to organize the distribution of terms.
Error-Proofing: Double-check by substituting values into the original and simplified expressions.
Concept Reinforcement: Practice with a variety of polynomials to reinforce understanding of the distributive property.