Lesson 2: Products of Polynomials

New Concept

This lesson expands on multiplication, starting with the First Law of Exponents ($a^m \cdot a^n = a^{m+n}$). We'll use this and the distributive law to multiply monomials, monomials by polynomials, and finally, any two or more polynomials together.

What’s next

Next, you’ll work through interactive examples and practice problems, applying these rules to multiply everything from simple monomials to more complex polynomials.

Property

To multiply two powers with the same base, we add the exponents and leave the base unchanged. In symbols,

Examples

• To simplify $z^4 \cdot z^5$, we keep the base $z$ and add the exponents: $z^{4+5} = z^9$.

Property

To multiply two monomials, rearrange the factors to group together the numerical coefficients and the powers of each base. Then, multiply the coefficients and use the first law of exponents for the variable factors.

Examples

• To multiply $(3a^2b)(4a^3b^4)$, we group and multiply: $(3 \cdot 4)(a^2 \cdot a^3)(b \cdot b^4) = 12a^5b^5$.

• The product of $(-6x^3y^2)(2x^5y)$ is found by multiplying coefficients and adding exponents of like bases: $(-6 \cdot 2)(x^3 \cdot x^5)(y^2 \cdot y) = -12x^8y^3$.

Property

To multiply a polynomial by a monomial, we use the distributive law. This means we multiply each term of the polynomial by the monomial.

Examples

• To multiply $4x(2x^2 - 5x + 3)$, distribute $4x$ to each term: $4x(2x^2) + 4x(-5x) + 4x(3) = 8x^3 - 20x^2 + 12x$.

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.

If a product contains a monomial factor, it is a good idea to multiply the polynomial factors together first, and save the monomial factor for last.

Examples

• To multiply $(x+5)(2x^2-x-3)$, we distribute term by term: $x(2x^2-x-3) + 5(2x^2-x-3) = 2x^3-x^2-3x+10x^2-5x-15 = 2x^3+9x^2-8x-15$.