homework #3 multiplying monomials

Multiplying Monomial

Multiplying monomials follow almost the same method as the multiplication of integers. To multiply monomials with polynomials(binomials or trinomials), we use the distributive property whereby the monomial is multiplied by each term in the other polynomial. The resulting polynomial is simplified further by adding or subtracting the identical terms.

What is Multiplying Monomial?

Multiplying monomial is a method for multiplying a monomial with other polynomials. A monomials is referred to as a type of polynomials with just one term, consisting of a variable and its coefficient. The monomial is multiplied with the individual terms of the polynomial and then simplified further to get the resultant polynomial. To multiply polynomials, the coefficient is multiplied with a coefficient, and the variable is multiplied with a variable.

Multiplying Monomial by a Monomial

When a monomial is multiplied by another monomial, their product will be a monomial only. A monomial is an expression that has only one term in it. The coefficients of the monomials are multiplied together and then the variables are multiplied. For example, the product of two monomials, say 2x and 2y is equal to 4xy.

In case, both the monomials have the same variables with the same exponents , then the laws of exponents are used.

Example: Multiply 2x 3 with 3x 2

• Step 1: First we will multiply the coefficients i.e., 2 × 3 = 6
• Step 2: Next, we will multiply the variables but in this case, the powers of both the variables will be added as per the rules of exponents i.e., x 3 × x 2 = x 5

The final answer is 2x 3 × 3x 2 = 6x 5

Multiplication of Three or More Monomials To find the product of more than two monomials, multiply the first two monomials, then multiply the product of these two by the third monomial. The same procedure is repeated for multiplying any number of monomials.

Multiplying Monomial by a Binomial

Binomials are also kind of polynomials consisting of only two terms. A binomial is defined as an algebraic expression consisting of two terms that are separated by the arithmetic signs, either the addition sign (+) or subtraction sign (-). When multiplying a monomial by a binomial, we follow the distributive law of multiplication. Let's take an example.

Example: Multiply (3x) (2x – 9)

Steps to solve (3x) (2x – 9):

• Step 1: Multiply the monomial with the first term of the binomial.

= (3x) * (2x) = 6x 2

• Step 2: Multiply the monomial with the second term of the binomial.

= (3 x ) *(-9) = -27 x

• Step 3: Write both the terms obtained in step 1 and step 2 together with their corresponding signs.

= 6x 2 - 27x

Multiplying Monomial by a Trinomial

Trinomials are a particular kind of polynomials consisting of three terms. A trinomial is defined as an algebraic expression consisting of three terms separated by arithmetic symbols/signs, either the addition sign (+) or subtraction sign (-). When multiplying a monomial by a trinomial, we follow the distributive law of multiplication. Let's take an example.

Example: Multiply (2x 2 ) (3x+9xy-6)

Steps to solve (2x 2 ) (3x+9xy-6):

• Step 1: Multiply the monomial with the first term of the trinomial.

= (2x 2 ) * (3x) = 6x 3

• Step 2: Multiply the monomial with the second term of the trinomial.

= (2 x 2 ) *(9xy) = 18x 2 y

• Step 3: Multiply the monomial with the third term of the trinomial.

= (2 x 2 ) *(-6) = -12 x 2

• Step 4: Write all the three terms together with their corresponding signs.

= 6x 3 + 18x 2 y -12 x 2

Related Articles

Check out these interesting articles to learn more about multiplying monomial and its related topics.

• Multiplying Polynomials Calculator
• Multiplying Binomials Calculator
• Polynomial Calculator

Multiplying Monomial Examples

Example 1: Using the multiplying monomial rule, multiply 11x and 2x.

Given: Monomials, 11x and 2x.

• Step 1: In the above monomials, the common variable is x. We will multiply the variable with the variable. Hence, we get x × x = x 2 .
• Step 2: Multiply the coefficients of both the monomials to get 11 × 2 = 22.

Thus, multiplying the polynomials 11x and 2x gives 22x 2 as the result.

Example 2: Multiply (2x)(4x 2 +7)

Given: A monomial and a binomial, 2x and 4x 2 +7.

• Step 1: Multiply the monomial 2x with the first term of the given binomial, which is 4x 2 . Hence, we get 2x × 4x 2 = 8x 2 .
• Step 2: Multiply the monomial 2x with the second term of the given binomial, which is 7. Hence, we get 2x × 7 = 14x.

Thus, multiplying the polynomials 2x and 4x 2 +7 gives 8x 2 + 14x as the result.

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Practice Questions on Multiplying Monomial

Faqs on multiplying monomial, what is multiplying monomial in algebra.

Multiplying Monomial is a method for multiplying a monomial with a polynomial. For multiplying monomials with polynomials(binomials or trinomials), we use the distributive property.

• Step 1: The monomial is multiplied by each of the terms in the other polynomial.
• Step 2: The resulting polynomial is then simplified.

What Is the Method for Multiplying a Monomial by a Monomial?

When multiplying monomials, follow the steps as given below:

• Step 1: Multiply the coefficients
• Step 2: Multiply the variables by adding the exponents.
• Step 3: Write the product so obtained

Note: When multiplying monomials with the same base, the base will remain the same, just add their exponents.

What Is the Product Rule for Multiplying Monomials?

As per the rule, multiply the coefficients first and then multiply the variables by adding the exponents. When monomials with the same base are multiplied, their exponents get added.

How To Solve Multiplying Monomials?

For multiplying a monomial with a polynomial, we usually follow distributive law.

• When multiplying two monomials, we rewrite the product as a single monomial using properties of multiplication and exponents.
• When multiplying a monomial with a binomial, we follow the distributive law of multiplication and we get a binomial as the product.
• When multiplying a monomial with a trinomial, we follow the distributive law of multiplication and we get a trinomial as the product.

What Are the Steps for Multiplying a Monomial by a Binomial?

When multiplying monomial by binomial, we follow the distributive law of multiplication.

• Multiply the monomial with the first term of binomial.
• Multiply the monomial with the second term of binomial.
• Write both the terms together with their corresponding signs.

Give the Steps for Multiplying a Monomial by a Trinomial?

When multiplying monomial by trinomial, we follow the distributive law of multiplication.

• Multiply the monomial with the first term of trinomial.
• Multiply the monomial with the second term of trinomial.
• Multiply the monomial with the third term of trinomial.
• Write all the three terms together with their corresponding signs.

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Multiplying monomials.

Multiplying monomials is a method of simplifying them by finding the product of their coefficients followed by the product of their variables.

For example, the product of two monomials, 5m and 7n, is:

5m ✕ 7n = (5 ✕ 7) ✕ (m ✕ n) = 35mn

Here, we observe that the product of monomials gives a monomial. Also, we usually write the variables in alphabetical order while writing the product. 

Now, let us multiply another two monomials, 2p and 3p 5 .

On multiplying the coefficients, we get 

(2 ✕ 3) = 6

On multiplying the variables, we get 

(p ✕ p 5 ), where both the variables have the same base and different exponents.

By applying the exponent rules (a m ✕ a n = a m+n ), we get

p 1+5 = p 6

Thus, the product is 6p 6 .

Find the product of 3a 2 b 3 and (4ab) 3

3a 2 b 3 ✕ (4ab) 3 Applying the exponent rules (a m ✕ b m = (ab) m ), we get 3a 2 b 3 ✕ 4 3 a 3 b 3 = 3a 2 b 3 ✕ 64a 3 b 3 Now, on multiplying the coefficients, we get (3 ✕ 64) = 192 On multiplying the variables, we get a 2 b 3 ✕ a 3 b 3 = (a 2 ✕ a 3 ) ✕ (b 3 ✕ b 3 ) As we know, a m ✕ a n = a m+n , the product of the variables is a 2+3 b 3+3 Thus, the product of the given monomials is obtained as 192a 5 b 6

With Three or More Monomials

To find the product of more than two monomials, we calculate the product of the first two monomials and then multiply it with the next monomial, and the process continues.

Let us multiply 3r 4 s 3 t, 2rs, and 5s 2 t 3 .

Now, on multiplying the coefficients, we get

(3 ✕ 2 ✕ 5) = 30

On multiplying the variables, we get

(r 4 s 3 t ✕ rs ✕ s 2 t 3 ) = (r 4 ✕ r) ✕ (s 3 ✕ s ✕ s 2 ) ✕ (t ✕ t 3 ) = r 1+4 s 3+1+2 t 1+3

Thus, the product is r 5 s 6 t 4 .

Multiply: (3x 3 ) 3 , 2x 2 t, and 5x 3 t 6 .

(3x 3 ) 3 ✕ 2x 2 t ✕ 5x 3 t 6 Applying the exponent rules (a m ✕ b m = (ab) m and (a m ) n = a mn ), we get 3 3 (x 3 ) 3 ✕ 2x 2 t ✕ 5x 3 t 6 = 27x 9 ✕ 2x 2 t ✕ 5x 3 t 6 Simplifying further, we get (27 ✕ 2 ✕ 5) ✕ (x 9 ✕ x 2 ✕ x 3 ) ✕ (t ✕ t 6 ) As we know, a m ✕ a n = a m+n , the product of the given monomials is 270x 9+2+3 t 1+6 = 270x 14 t 7 .

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Multiplying monomials, learning outcomes.

• Use the power and product properties of exponents to multiply monomials
• Use the power and product properties of exponents to simplify monomials

Properties of Exponents

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Course: precalculus   >   unit 4, multiplying & dividing rational expressions: monomials.

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1.2.4: Powers of Monomials and Binomials

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Learning Objectives

By the end of this section, you will be able to: 

• Expand a positive integer power of monomial and binomial expressions
• Identify coefficients of terms of a positive integer powers of a binomial expression

Be Prepared

Before we get started, take this readiness quiz.

1. Expand \((2x-3)^2 \).

2. Simplify \((3\cdot 5\).

3. Evaluate \((5x^3\) at \(x=2\).

Powers of Monomials

Now let’s look at an exponential expression that contains a power raised to a power. Let's see if we can discover a general property.

Notice the 6 is the product of the exponents, 2 and 3. We see that \((x^2)^3\) is \(x^{2\cdot 3}\) or \(x^6\). We can also see that

In this example we multiplied the exponents.

We can check various examples to see that this leads us to the Power Property for Positive Integer Exponents.

Power Property for Integer Exponents

If \(a\) is a real number and \(m\) and \(n\) are positive integers, then

\[(a^m)^n=a^{mn}. \nonumber \]

To raise a power to a power, multiply the exponents.

Example \(\PageIndex{1}\)

Simplify each expression:

a. \((y^5)^9\)

b. \((4^{4})^7\)

c. \((y^3)^6(y^5)^4\)

Try It \(\PageIndex{2}\)

a. \((b^7)^5\)

b. \((5^4)^{3}\)

c. \((a^4)^5(a^7)^4\)

a. \(b^{35}\)

b. \(5^{12}\)

c. \(a^{48}\)

Try It \(\PageIndex{3}\)

a. \((z^6)^9\)

b. \((3^{7})^7\)

c. \((q^4)^5(q^3)^3\)

a. \(z^{54}\)

b. \(3^{49}\)

c. \(q^{29}\)

We will now look at an expression containing a product that is raised to a power. Can we find this pattern?

Notice that each factor was raised to the power and \((2x)^3\) is \(2^3x^3\).

The exponent applies to each of the factors! We can say that the exponent distributes over multiplication. If we were to check various examples with exponents we would find the same pattern emerges. This leads to the Product to a Power Property for Postive Integer Exponents.

Product to a Power Property for Integer Exponents

If \(a\) and \(b\) are real numbers and \(m\) is a positive integer, then

\[(ab)^m=a^mb^m \nonumber. \]

To raise a product to a power, raise each factor to that power.

Example \(\PageIndex{4}\)

a. \((−3mn)^3\)

b. \((6k^3)^{2}\)

c. \((5x^{3})^2\)

Try It \(\PageIndex{5}\)

a. \((2wx)^5\)

b. \((2b^3)^{4}\)

c. \((8a^{4})^2\)

a. \(32w^5x^5\)

b. \(16b^{12}\)

c. \(64a^8\)

Try It \(\PageIndex{6}\)

a. \((−3y)^3\)

b. \((−4x^4)^{2}\)

c. \((2c^{4})^3\)

a. \(−27y^3\)

b. \(16x^8\)

c. \(8c^{12}\)

• The Binomial Theorem

In this section we consider powers of binomial expressions like

$$(x+y)^5, (x+3)^4, \text{ or }(2x-3)^{10}.\nonumber$$

These are polynomials with degree equal to the exponent.

Let's consider

$$(x+y)^3=(x+y)(x+y)(x+y).\nonumber$$

To distribute, we take one term from each factor and multiply.  We repeat for all possible choices then add the results.

$$(x+y)^3=(x+y)(x+y)(x+y)=xxx+xxy+xyx+yxx+xyy+yxy+yyx+yyy=x^3+3x^2y+3xy^2+y^3.\nonumber$$

If we consider the coefficient of the \(x^2y\) term, we see it is the number of ways we can choose two \(x\)'s and one \(y\).  Similarly, for the other coefficients.

It turns out these are well known numbers and we have

The Binomial Theorem \(\PageIndex{7}\)

 \[(x+y)^n=a_0x^n+a_1x^{n-1}y+\cdots+a_{n-1}xy^{n-1}+a_ny^n,\nonumber\]

where the coefficients come from the \(n\)th row (counting from \(0\) of the Pascal's triangle):

See  Pascal's triangle - Wikipedia   -- The first paragraph.  The image above is from the same.

To get from one row to the next you add the two numbers above (or 1 in the case of the first and last number).  For example on the 4th row (counting from zero), above the 4 are 3 and 1, above the 6 is 3 and 3, and the ends have only 1 above.

For example,

\[\begin{align*}(x+2)^4&=x^4+4x^3\cdot 2+6x^2\cdot 2^2+4 x \cdot 2^3+2^4\\ &=x^4+8x^3+24 x^2+32 x+16.\end{align*}\]

Note that the numbers in Pascal's triangle are called binomial coefficients.  The details are beyond the scope of the book but more information and many applications can be found in the Wikipedia article on the topic.

Example \(\PageIndex{8}\)

Find the coefficient of \(x^3\) in the expression \((2x-3)^6\).

Note that the exponent on \(x\) is 3.  We find the 4th number in the 7th line of Pascal's triangle (this is because with Pascal's triangle the counting starts at 0).  Thus, we see it is 20.

So the relevant term is \(20 (2x)^3(-3)^3\) (noting that the exponents add to \(6\)).  Therefore, the coefficient of \(x^3\) is 

\[20\cdot 2^3\cdot(-3)^3=-4320.\nonumber\]

Try It \(\PageIndex{9}\)

Find the coefficient of

a. \(x^4\) in \((2-x)^7\)

b. \(xy^2\) in \((x-3y)^3\)

c. \(x^4y^3\) in \((2y-x^2)^5\)

Writing Exercises \(\PageIndex{10}\)

• Explain by writing out the full meaning why \((2^3)^4\).
• Why is \((2^3)^4=(2^4)^3\)?
• Verify the expansion of \((x+1)^4\) by evaluating at \(x=1\).

Exit Problem

• Simplify \((3x^4)^3\) and check by evaluating this expression and your simplification at \(x=2\).
• What is the coefficient of \(x^5y^2\) in \(2x-y)^7\)

Key Concepts

• Power of a monomial
• Power of a binomial
• Pascal's triangle

Multiplying and Dividing Monomials Worksheet and Answer Key

Students will practice multiplying and dividing monomials.

Example Questions

Directions: Multiplying the monomials below.

Other Details

This is a 4 part worksheet:

• Part I Model Problems
• Part II Practice
• Part III Challenge Problems
• Part IV Answer Key
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• Multiplying Monomials

Multiplying monomials requires that you understand and can apply the laws of exponents. Let's quickly review the laws of exponents that will be needed.

Laws of Exponents

• Multiplying Powers with the Same Base - Add the exponents. For example:            ( a 3 ) (a 2 ) = a 5
• Power of a Power - Multiply the exponents. For example: (a 3 ) 5 = a 15
• Power of a Product -find the power of each factor and multiply. For example: (3a) 2 = 9a 2

When you multiply monomials, you will need to perform two steps:

• Multiply the coefficients (Numbers)
• Multiply the variables. (Use the laws of exponents when necessary)

Let's look at a few examples.

Each line of these examples shows a different step.

I broke it down step by step for you to see the exact process. You will be able to do a lot of this work mentally, which I will show you later on.

I wanted you to understand each step, that's why these explanations are so long.

This first example shows how you use multiplying powers with the same base in order to multiply monomials.

Example 1: Multiplying Powers with the Same Base

This second example will utilize the Power of a Product property in order to simplify the expression.

Example 2: Using Power of a Product

One more example, but this time I'm not going to show each individual step. As you master this skill, this is the way in which you will multiply monomials. It's actually pretty quick!

Example 3: Multiplying Monomials

For further help with monomials, you may want to visit our next page on dividing monomials.

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Adding, Multiplying, and Subtracting Monomials Worksheets

How to Add, Multiply, and Subtract Monomials? You must have come across several algebraic expressions. Some have one variable; some have more than two and some variables are even raised to the powers. Well, there are different types of these algebraic expressions. But for now, we are going to investigate one such expression. Monomial expression - Let's start with the simplest expression, a monomial. An algebraic expression that consists of one term is called monomial. You can easily remember this by learning that 'mono' in monomial means one. So, a monomial is a one-term algebraic expression. Moving on to learning the basic mathematical operations with monomial. Adding monomials - Two or more monomials can be added only when they are like terms. Like terms are those expressions that have identical variables and exponents. They may or may not have different coefficients. While adding monomials, you only add the coefficients and the variables stay the same . For example; 2 x 2y + 3 x 2y = 5 x 2y. Notice only the coefficients have added up, while the variables and exponents remained unchanged. Subtracting monomials - The subtraction of monomial follows the same rules as with addition. The coefficients are subtracted from, each other while the variables are not changed For example; 3 x 2y – 2 x 2y = 1 x 2y. The coefficients are subtracted, but the variables stay the same. Multiplying monomials - Multiplication of monomial is different from addition and subtraction. In multiplication, both coefficient and variables are multiplied. With variable multiplication, we use the rule of power. That is, the exponents of bases are added. For example; (5x)(5 x 2) = 5.5 (x 1 + 2) = 25 x 3

Aligned Standard: **Related to HSA-APR.A.1**

• Adding, Subtracting Monomials Lesson - It is a sunny day for monomials. You can use this section as a warm-up for polynomials.
• Multiplying Monomials Lesson - Remember that when you multiply exponents, you basically add the values.
• Adding & Subtracting Monomials Worksheet 1 - We start off simple and throw two or three slightly difficult problems in there.
• Multiplying Monomials Worksheet 3 - The difficulty level is slightly higher here than the sheets below.
• Adding & Subtracting Monomials Homework - Why not send them home with a brief reminder of what you worked on today.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

These are great for extra work that make all the difference in your week.

• Homework 2 - Adding & Subtracting: Find each sum or difference.
• Homework 3 - Multiplication: Find each product.
• Homework 4 - More Multiplying: Determine the end values after multiplying monomials.

Practice Worksheets

These should print out fully spaced and with answer keys.

• Add & Subtract Worksheet 2 - Find each sum or difference.
• Multiplying Practice 4 - The first problem has the variable "i" as in "Irene".

Math Skill Quizzes

A quiz on each of the skills. Of course sums and differences are together.

• Adding & Subtracting Quiz 1 - A helpful quiz sheet.
• Multiplying Quiz 2 - Don't get thrown when a variable has no number in front of it, it is equal to one.

When Do We Apply These Operations?

It should be noted that we can only add or subtract monomials if they contain like terms. This means that they contain the same variables and corresponding exponents. The coefficients on these like terms may be different and that is where our operators indicate what to do with them. When we are performing multiplication operations it is important to note that we should be focused on the coefficients first and then we process the variables by finding the sum of the exponents. This will help in many different situations where are attempting to simplify expressions to give us something that easier to work with.

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