Section 1
General Polynomial Multiplication
Property
To find the product of two polynomials, multiply each term of the first polynomial by each term of the second polynomial, then combine any like terms.
In this Grade 4 AMC Math lesson from AoPS: Introduction to Algebra, students learn how to multiply polynomials using the distributive property and an organized column method modeled after integer multiplication. The lesson covers key concepts including degree of a product, leading terms, constant terms, and monic polynomials, with worked examples such as expanding expressions like (3y² − 2y + 3)(y³ − 2y² + y − 7). Students also explore why the product of any two polynomials must itself be a polynomial and how the degrees of the factors relate to the degree of the product.
Section 1
General Polynomial Multiplication
To find the product of two polynomials, multiply each term of the first polynomial by each term of the second polynomial, then combine any like terms.
Section 2
Multiply a Trinomial by a Binomial
We are now ready to multiply a trinomial by a binomial.
Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.
For the Vertical Method, it is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.
FOIL doesn't work here because there are more than four terms to multiply. Instead, use the Distributive Property or the Vertical Method. Both ensure that every term in the first polynomial multiplies every term in the second.
Section 3
The vertical method
The Vertical Method for multiplying polynomials is analogous to multiplying whole numbers. Write one polynomial above the other. Multiply the top polynomial by each term of the bottom polynomial, creating partial products. Align like terms in columns and add the partial products to get the final answer.
Expand to review the lesson summary and core properties.
Section 1
General Polynomial Multiplication
To find the product of two polynomials, multiply each term of the first polynomial by each term of the second polynomial, then combine any like terms.
Section 2
Multiply a Trinomial by a Binomial
We are now ready to multiply a trinomial by a binomial.
Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.
For the Vertical Method, it is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.
FOIL doesn't work here because there are more than four terms to multiply. Instead, use the Distributive Property or the Vertical Method. Both ensure that every term in the first polynomial multiplies every term in the second.
Section 3
The vertical method
The Vertical Method for multiplying polynomials is analogous to multiplying whole numbers. Write one polynomial above the other. Multiply the top polynomial by each term of the bottom polynomial, creating partial products. Align like terms in columns and add the partial products to get the final answer.