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Multiplying Terms using Distribution

Multiplying Terms using Distribution

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In this lesson, students will learn how to multiply terms by using the distribution method and rules of exponents.

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Tutorial
This tutorial covers how to multiply terms using distribution, through the definition and discussion of:
  1. Monomials, Binomials, and Polynomials
  2. The Distributive Property
  3. Multiplying Monomials and Binomials
  4. Multiplying Monomials and Polynomials


1. Monomials, Binomials, and Polynomials

A monomial is an exponential expression that consists of one term with non-negative integer exponents—“mono” meaning “one.”

Monomial
An exponential expression with non-negative integer exponents

Following is an example of a monomial. In this term, 4 is the coefficient, x is the base, and 5 is the exponent.

File:1173-mono1.PNG

A binomial is an expression that consists of two monomial terms—“bi” meaning “two.”

Binomial
An expression containing two monomial terms

Here is an example of a binomial:

File:1174-mono2.PNG

Lastly, a polynomial is an expression that consists of two or more monomial terms—“poly” meaning “many.”

Polynomial
An expression containing two or more monomial terms

Here is an example of a polynomial. Expressions should always be simplified by combining like terms if possible, so in this polynomial, you can combine the like terms 4a and 2a.

File:1175-mono3.PNG


2. The Distributive Property

The distributive property is where the quantity that is outside of the parentheses is multiplied or distributed into every term inside the parentheses.

a left parenthesis b plus c right parenthesis equals a b plus a c

Suppose you want to simplify the following expression. You would distribute by multiplying the 7 to both the x and the -4 in the parentheses.

table attributes columnalign left end attributes row cell 7 left parenthesis x minus 4 right parenthesis equals end cell row cell 7 x minus 28 end cell end table


3. Multiplying Monomials and Binomials

When multiplying a monomial by a binomial, the entire monomial is distributed, including the coefficients and the variable powers. Suppose you want to multiply:

4 m cubed left parenthesis 5 m squared plus 2 m right parenthesis

Because your variable bases are the same, you can use the product property for exponents to add your exponents together when multiplying the variable powers together.

To multiply, you will use the distributive property to multiply 4m^3 by the terms within the parentheses. When you multiply 4m^3 by 5m^2 , you can use the commutative property of multiplication to rewrite the product by grouping your coefficients together and your variable powers together. Similarly, when you multiply 4m^3 by 2m, you can again use the commutative property of multiplication to rewrite your product.

left parenthesis 4 times 5 right parenthesis left parenthesis m cubed times m squared right parenthesis plus left parenthesis 4 times 2 right parenthesis left parenthesis m cubed times m right parenthesis

To simplify your first term, multiply your coefficients, 4 and 5, which equals 20. Next, use the product property of exponents to add your exponents together.

table attributes columnalign left end attributes row cell left parenthesis 4 times 5 right parenthesis left parenthesis m cubed times m squared right parenthesis equals end cell row cell 20 m to the power of 5 end cell end table

Simplifying your second term, multiply your coefficients, 4 times 2, which equals 8. Note that your variable m has no written exponent, which means that it has an implied exponent of 1. Therefore, adding your exponents together provides:

table attributes columnalign left end attributes row cell left parenthesis 4 times 2 right parenthesis left parenthesis m cubed times m right parenthesis equals end cell row cell 8 m to the power of 4 end cell end table

Bringing your two terms back together, your final expression is in standard form because the term with the highest exponent power is written first, followed by the term with the next highest exponent power, and so on.

20 m to the power of 5 plus 8 m to the power of 4
You can also call standard form of a polynomial “descending order.”

Now, using what you’ve learned, try multiplying the following monomial and binomial terms.
negative 2 x squared left parenthesis 3 x to the power of 6 minus x right parenthesis
You need to multiply -2x^2 by both terms in the parentheses, so begin by multiplying it by the first term in the parentheses.
negative 2 x squared times 3 x to the power of 6 equals negative 6 x to the power of 8
Next, multiply -2x^2 times your second term, -x. Note that the x here has an implied coefficient of -1.
negative 2 x squared times negative 1 x equals 2 x cubed
Bringing the two terms back together, here is your resulting expression:
negative 6 x to the power of 8 plus 2 x cubed

It is in standard form or descending order, so this is your final answer.


4. Multiplying Monomials and Polynomials

The same rules apply when multiplying monomials with polynomials. Consider the example below. You need to distribute the 3 on the outside of the parentheses to all three terms within the parentheses. Notice that the 3 does not have any variable bases or exponents, so you only need to multiply the coefficients. Also, remember that there is an implied -1 being multiplied by the b here.

table attributes columnalign left end attributes row cell 3 left parenthesis 2 a squared plus a cubed b squared minus b right parenthesis equals end cell row cell 6 a squared plus 3 a cubed b squared minus 3 b end cell end table

Now you must reorder your terms so that they are in descending order, so you’ll need to switch the first and second terms to arrive at your final answer.

3 a cubed b squared plus 6 a squared minus 3 b

For this last example, you’ll need to multiply the term on the outside by each of the three terms inside the parentheses. Note that the outside term contains variables, coefficients, and exponents.

7 x squared y cubed left parenthesis 2 x to the power of 5 minus 5 x squared y squared plus 3 y cubed right parenthesis

Begin by multiplying the outside term by the first term in the parentheses. 7 times 2 is 14, and you can also add the exponents of x^2 and x^5 because their bases are both x; y^3 stays unchanged. Next, multiply the outside term by your second and third term in the parentheses. Your expression is in standard form or descending order, so this is your final answer:

table attributes columnalign left end attributes row cell 7 x squared y cubed left parenthesis 2 x to the power of 5 minus 5 x squared y squared plus 3 y cubed right parenthesis equals end cell row cell 14 x to the power of 7 y cubed minus 35 x to the power of 4 y to the power of 5 plus 21 x squared y to the power of 6 end cell end table

Today your learned the definitions of monomials, binomials, and polynomials. You also reviewed the distributive property, and learned how to apply it when multiplying monomials by binomials and polynomials, keeping in mind that the entire monomial is distributed, including coefficients and variable powers.

Source: This work is adapted from Sophia author Colleen Atakpu.

TERMS TO KNOW
  • Monomial

    An exponential expression with non–negative integer exponents.

  • Binomial

    An expression containing two monomial terms.

  • Polynomial

    An expression containing two or more monomial terms.