To multiply and divide algebraic terms, it’s crucial that you known the laws of exponents. This is why we started started with addition and subtraction (which don’t require the laws), followed by the exponents laws, and now leading to this.
Multiplying a Monomial by a Monomial
Recall that a monomial is an algebraic expression consisting of a single term. Shown below are a few ways to represents two factors (b and d) being multiplied:
To multiply two or monomials, you apply the laws of exponents to individual factors making up that term. For example, consider multiplying the monomial 2x³ by 5x. 2x³ consists of two factors, 2 and x³, while 5x consists of the factors 5 and x. To multiply these, we multiply 2 with 5, and x³ with x; the latter is where we apply the product rule taught last lesson. One of five ways to represent this calculation is shown below:
The other ways to represent it are shown underneath:
A visual demonstration of several other examples ranging in complexity and difficulty are shown in the video:
Multiplying a Monomial by a Binomial
To multiply a monomial by a binomial, you’ll need to know the distributive property of multiplication summarized below:
Notice how the monomial a gets multiplied to both the b and c terms within the parentheses – this process is called expanding. Once you’re done expanding, remove the parentheses as it’s no longer needed; after all, expanding is analogous to degrouping a set of terms. Try expanding the expression below on your own before watching a tutorial:
The first question shows three examples of varying difficulty. Question 2 caters to multiplying polynomials; you should hold off on watching that part, because it’ll be introduced later in this lesson.
Multiplying a Binomial by a Binomial
To multiply two binomials side-by-side, such as (a + b)(c + d), the FOIL method is used to expand the two polynomials.
FOIL is an acronym for First, Outer, Inner, Last, and is a mnemonic for the standard method of multiplying two binomials. The video below shows how to expand and simplify the two binomials listed:
- (7x − 5)(4x − 3)
- (7x + 6y)(3x − y)
Multiplying a Polynomial by a Polynomial
Multiplying polynomials requires that you understand how to expand binomials first. One thing to remember is that if you’re multiplying several polynomials embedded inside parentheses, always focus on two at a time. The video below consists of two questions, only question 2 relates to the multiplication of polynomials.
Dividing a Polynomial by a Monomial
The quotient rule of the exponent laws is one of two prerequisites to learning how to divide a polynomial by a monomial. The second is understanding fraction notation. Take for example the fraction:
You could rewrite the denominator (our monomial) so that it’s underneath both terms in the numerator:
Then each term can be individually reduced using the quotient rule. The examples the in the video below demonstrate how it’s done.
Given all the ideas introduced in this lesson, see if you expand and simplify the examples listed below.
- −2x(3x + 1) −3(4x − 3)
- 2(5m − 2)(m + 3) − m(m − 1)²
The solutions are also provided in the video.