Problem 204 Multiply the following binomials... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 204

Multiply the following binomials. Use any method. $$(r+s)(3 r+2 s)$$

Short Answer

Expert verified

3r^2 + 5rs + 2s^2

Step by step solution

- Distribute the First Term

Multiply the first term in the first binomial $r$ by each term in the second binomial $3r + 2s$. \[ r \times 3r = 3r^2 \] \[ r \times 2s = 2rs \]

- Distribute the Second Term

Multiply the second term in the first binomial $s$ by each term in the second binomial $3r + 2s$. \[ s \times 3r = 3rs \] \[ s \times 2s = 2s^2 \]

- Combine Like Terms

Add all the products obtained in the previous steps to get the final expanded form. \[ 3r^2 + 2rs + 3rs + 2s^2 \] Combine the like terms: \[ 3r^2 + 5rs + 2s^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distribution Method

The distribution method is a systematic way to multiply binomials. It involves multiplying each term in one binomial by each term in the second binomial, one at a time.

Let's break this down using the given example:

(r + s)(3r + 2s)

First, you distribute the first term of the first binomial: - r 脳 3r = 3r虏 - r 脳 2s = 2rs Then, you distribute the second term of the first binomial: - s 脳 3r = 3rs - s 脳 2s = 2s虏
By carefully carrying out these multiplications, we make sure no term is missed.

Combining Like Terms

After distributing, you'll get a sum of several terms, some of which might be similar. These similar terms are called 'like terms'. Combining like terms means adding or subtracting the coefficients of terms that have the same variables raised to the same powers.

In our problem, after distribution, we get:
3r虏 + 2rs + 3rs + 2s虏
Here, 2rs and 3rs are like terms because they both include the product rs.
To combine them, add their coefficients: 2 and 3:
2rs + 3rs = 5rs
Putting it all together, we have:
3r虏 + 5rs + 2s虏

Binomial Expansion

The binomial expansion involves spreading out a binomial expression into a polynomial. This step-by-step multiplication resulted in a quadratic polynomial.

For the binomials (r + s) and (3r + 2s), the expanded form is:
(r + s)(3r + 2s) = 3r虏 + 5rs + 2s虏
This polynomial expression is the binomial expansion of the original product.
Notice how each term in the binomials contributes to the final expanded form. Through the distribution method, we ensure that every term in one binomial is multiplied by every term in the other.

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91影视

Multiply the following binomials. Use any method. $$(r+s)(3 r+2 s)$$

Short Answer

Step by step solution

- Distribute the First Term

- Distribute the Second Term

- Combine Like Terms

Key Concepts

Distribution Method

Combining Like Terms

Binomial Expansion

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