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:

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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:

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The other ways to represent it are shown underneath:

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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:

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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:

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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.

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*FOIL* is an acronym for **F**irst, **O**uter, **I**nner, **L**ast, 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:

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You could rewrite the **denominator** (our monomial) so that it’s underneath both terms in the **numerator**:

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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.