What is the Product of a Monomial and a Binomial?

S3-SA1-0243

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The product of a monomial and a binomial is what you get when you multiply a single-term algebraic expression (monomial) by a two-term algebraic expression (binomial). You use the distributive property to multiply each term in the binomial by the monomial.

Simple Example

Quick Example

Imagine you are buying 2 packets of biscuits and each packet has 3 cream biscuits and 4 plain biscuits. To find the total number of biscuits, you multiply 2 by (3 cream + 4 plain). Similarly, to find the product of 2x and (3y + 4z), you multiply 2x by 3y and 2x by 4z separately, then add the results.

Worked Example

Step-by-Step

Let's find the product of 3a and (2b + 5c).

Step 1: Identify the monomial and the binomial. Here, 3a is the monomial and (2b + 5c) is the binomial.

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Step 2: Apply the distributive property. This means you multiply the monomial (3a) by the first term of the binomial (2b).
3a * 2b = 6ab

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Step 3: Now, multiply the monomial (3a) by the second term of the binomial (5c).
3a * 5c = 15ac

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Step 4: Add the results from Step 2 and Step 3.
6ab + 15ac

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Answer: The product of 3a and (2b + 5c) is 6ab + 15ac.

Why It Matters

Understanding this helps in solving complex problems in physics, like calculating distances or forces. Engineers use this concept to design structures and calculate material needs. Even in computer science, this helps in understanding algorithms and data structures, which are vital for creating apps and software.

Common Mistakes

MISTAKE: Multiplying the monomial only by the first term of the binomial. For example, 2x(y + 3) = 2xy. | CORRECTION: Remember to multiply the monomial by BOTH terms inside the binomial. So, 2x(y + 3) = 2xy + 6x.

MISTAKE: Making errors with signs, especially negative signs. For example, -3(x - 2) = -3x - 6. | CORRECTION: Always multiply the signs correctly. A negative times a negative is a positive. So, -3(x - 2) = -3x + 6.

MISTAKE: Incorrectly combining terms after multiplication. For example, 4(a + 2b) = 4a + 8b, then trying to add 4a and 8b to get 12ab. | CORRECTION: Only combine like terms. 4a and 8b are not like terms, so the expression 4a + 8b cannot be simplified further.

Practice Questions

Try It Yourself

QUESTION: Find the product of 5 and (x + 7). | ANSWER: 5x + 35

QUESTION: Multiply 4y by (2y - 3). | ANSWER: 8y^2 - 12y

QUESTION: Simplify the expression: -2a(3a + 4b). | ANSWER: -6a^2 - 8ab

MCQ

Quick Quiz

What is the product of 6p and (p + 5)?

6p + 30

6p^2 + 5

6p^2 + 30p

30p^2

The Correct Answer Is:

To find the product, multiply 6p by p to get 6p^2, and then multiply 6p by 5 to get 30p. Adding these results gives 6p^2 + 30p.

Real World Connection

In the Real World

When calculating the total cost for buying different items from a grocery store like Big Bazaar, if you buy 'x' quantity of one item and 'y' quantity of another, and there's a discount 'd' applied per item, you might use expressions like d(x + y) to quickly calculate the total discount, which involves multiplying a monomial by a binomial.

Key Vocabulary

Key Terms

MONOMIAL: An algebraic expression with only one term, like 3x or 5. | BINOMIAL: An algebraic expression with two terms, like (x + 2) or (2a - 3b). | PRODUCT: The result obtained when two or more numbers or expressions are multiplied together. | DISTRIBUTIVE PROPERTY: A property that states multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.

What's Next

What to Learn Next

Great job learning about multiplying a monomial by a binomial! Next, you can explore 'What is the Product of Two Binomials?'. This builds on what you've learned and will help you solve even more complex algebraic problems.