8.4 - Multiplying a Monomial by a Polynomial
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Monomials and Polynomials
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Today we're going to discuss how to multiply a monomial by a polynomial. Remember, a monomial is an expression with just one term. Can anyone give me an example of a monomial?
4x is a monomial!
That's correct! Now, a polynomial can have one or more terms. Who can give me an example?
How about x² - 3x + 5?
Great job! Now, we will focus on multiplying a monomial with a polynomial, such as the binomial. What do we remember about the distributive law?
It means we multiply each term in the parentheses separately!
Exactly! Let's remember to distribute with an acronym: D for Distribute, E for Every term. Now, let’s see an example.
Multiplying a Monomial by a Binomial
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Let’s compute 3x × (5y + 2). Using the distributive law: what will be our first step?
We multiply 3x by both 5y and 2!
That's right! So we get: 15xy + 6x. Can anyone simplify: 7x multiplied by (2a - 4)?
That would be 14ax - 28!
Fantastic! Let's summarize: Remember to always apply our D.E. method. Distribute across every term.
Multiplying a Monomial by a Trinomial
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Next, let’s multiply a monomial by a trinomial. For instance, 3p × (4p² + 5p + 7). What’s our first step?
We’ll multiply 3p by each term in the trinomial!
Correct! So, we get 12p³ + 15p² + 21p. Why do we have three terms?
Because there are three terms in the trinomial!
Exactly! Great job! Remember to look out for combining like terms when multiplying!
Application and Practice
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
To apply these skills, let’s think of practical examples. If I say the length of a rectangle is (3x) and its width is (4y + 2), what is the area?
We would multiply 3x by (4y + 2) to get 12xy + 6x!
Great! Let's practice multiplying some more. What is the product of 5(a + 3)?
It's 5a + 15.
Fantastic! I want everyone to practice these problems and remember our D.E. method.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, students learn how to multiply a monomial by a polynomial, covering both binomials and trinomials. The distributive law is introduced as a fundamental concept to simplify calculations, and examples illustrate the step-by-step process and applications.
Detailed
Detailed Summary
This section, titled "Multiplying a Monomial by a Polynomial," outlines the method of multiplying monomials with polynomials, highlighting binomials and trinomials as key categories of polynomial expressions.
The multiplication of a monomial by a binomial is illustrated using examples, such as multiplying 3x by the binomial (5y + 2). Using the distributive law, the multiplication is expressed as:
- Distributive Law: 3x × (5y + 2) = (3x × 5y) + (3x × 2) = 15xy + 6x.
The section further explores multiplying a monomial by a trinomial through an example of 3p × (4p² + 5p + 7), resulting in:
- (3p × 4p²) + (3p × 5p) + (3p × 7) = 12p³ + 15p² + 21p.
Prominent throughout is the emphasis on the distributive law, a principle that allows the simplification of expressions by distributing the monomial across each term in the polynomial. The section also underscores the importance of recognizing like terms when combining results after multiplication. By the end of this section, students will have established a clear understanding of multiplying monomials by polynomials, thereby strengthening their foundation for more complex algebraic operations.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Definition of Polynomials
Chapter 1 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
An expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on. In general, an expression containing one or more terms with non-zero coefficients (with variables having non-negative integers as exponents) is called a polynomial.
Detailed Explanation
In mathematics, polynomials are expressions made up of variables and coefficients. A single term is called a monomial, two terms make a binomial, and three terms create a trinomial. For instance, '3x + 4' is a binomial, while '2x^2 + 3x + 5' is a trinomial.
Examples & Analogies
Think of polynomials like a recipe: a monomial is a single ingredient, a binomial combines two ingredients, and a trinomial adds a third ingredient. Just as you can combine various ingredients to create different dishes, you can combine terms in mathematics to create different expressions.
Multiplying a Monomial by a Binomial
Chapter 2 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Let us multiply the monomial 3x by the binomial 5y + 2, i.e., find 3x × (5y + 2) = ? Recall that 3x and (5y + 2) represent numbers. Therefore, using the distributive law, 3x × (5y + 2) = (3x × 5y) + (3x × 2) = 15xy + 6x.
Detailed Explanation
Multiplying a monomial by a binomial involves distributing the monomial across each term in the binomial. In our example, we take 3x and multiply it by both 5y and 2 separately. The results of these multiplications (15xy and 6x) are then added together to get the final product of 15xy + 6x.
Examples & Analogies
Imagine you are packaging boxes of fruit. If you have 3 boxes (the monomial) and each box has 5 apples and 2 oranges (the binomial), you would calculate the total number of fruits you have by multiplying the number of boxes by the amount of each type of fruit.
Multiplying a Monomial by a Trinomial
Chapter 3 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Consider 3p × (4p² + 5p + 7). As in the earlier case, we use the distributive law; 3p × (4p² + 5p + 7) = (3p × 4p²) + (3p × 5p) + (3p × 7) = 12p³ + 15p² + 21p.
Detailed Explanation
When multiplying a monomial by a trinomial, we apply the same distributive law by multiplying the monomial by each term of the trinomial separately. Here, we take 3p and multiply it by 4p², 5p, and 7. This gives us three products, which we then add together to yield the final answer of 12p³ + 15p² + 21p.
Examples & Analogies
Consider a gardener with a set of plants. If the gardener has 3 pots of plants, and each pot has 4 potted flowers, 5 potted herbs, and 7 potted vegetables (this is the trinomial), the gardener can find out the total number of each type by multiplying the number of pots by each type of plant separately.
Using Distributive Law in Examples
Chapter 4 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
For example: (–3x) × (–5y + 2) = (–3x) × (–5y) + (–3x) × (2) = 15xy – 6x.
Detailed Explanation
Using the distributive law can also apply to negative numbers. Here, when we multiply –3x by –5y, the product is positive, resulting in 15xy. When we multiply –3x by 2, we get –6x. So, the entire expression simplifies to 15xy – 6x.
Examples & Analogies
If you owe money (represented by –3x) but receive payments (represented by a positive outcome of the binomial) for two different types of loans, you can calculate new totals. The distribution ensures you accurately account for each type of transaction.
Terminology: Binomials and Trinomials
Chapter 5 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
What about a binomial × monomial? For example, (5y + 2) × 3x = ? We may use commutative law as: 7 × 3 = 3 × 7
Detailed Explanation
The commutative law states that the order in which two numbers are multiplied does not change the product. So whether we calculate (5y + 2) × 3x or 3x × (5y + 2), we will arrive at the same result, 15xy + 6x. This highlights the flexibility in multiplication – we can rearrange terms as needed.
Examples & Analogies
Imagine a light-weight box filled with 5 bags of flour and two bags of sugar. Whether we multiply the amount of flour first and then add sugar, or combine both types simultaneously, will yield the same total weight of the goods you're carrying.
Key Concepts
-
Distributive Law: A method for multiplying a single term by each term in a polynomial.
-
Monomial: An algebraic expression consisting of exactly one term.
-
Polynomial: An algebraic expression containing one or more terms.
-
Binomial: A polynomial with two terms.
-
Trinomial: A polynomial with three terms.
Examples & Applications
3x × (5y + 2) = 15xy + 6x.
3p × (4p² + 5p + 7) = 12p³ + 15p² + 21p.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
When you multiply, don't forget the rule, Distribute each term, that's the golden tool!
Stories
Once upon a time, there was a wise owl named 'Distributive.' He taught the students to share everything equally, so when multiplying, they learned to distribute their values across all terms in a polynomial.
Memory Tools
D.E. = Distribute Every term! A reminder to always distribute correctly!
Acronyms
M.P.P. = Monomial x Polynomial = Product in Polynomial form.
Flash Cards
Glossary
- Monomial
An algebraic expression containing one term.
- Polynomial
An algebraic expression containing one or more terms.
- Binomial
A polynomial that contains two terms.
- Trinomial
A polynomial that contains three terms.
- Distributive Law
A property of multiplication that states a(b + c) = ab + ac.
Reference links
Supplementary resources to enhance your learning experience.