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Chapter 10: Problem 195

Multiply the following binomials. Use any method. $$(6 n+5)(n+1)$$

Short Answer

Expert verified
The product is \( 6n^2 + 11n + 5 \).

Step by step solution

01

- Distribute Each Term

Start by distributing each term in the first binomial to every term in the second binomial. Multiply the first term of the first binomial by each term in the second binomial:\[ 6n \times (n + 1) \]
02

- Multiply First Term

Multiply the first term of each binomial:\[ 6n \times n = 6n^2 \]and then\[ 6n \times 1 = 6n \]
03

- Multiply Second Term

Now multiply the second term of the first binomial by each term in the second binomial:\[ 5 \times n = 5n \]and then\[ 5 \times 1 = 5 \]
04

- Combine Like Terms

Add the results from each step together:\[ 6n^2 + 6n + 5n + 5 \]Now combine the like terms (the terms with the same variable to the same power):\[ 6n^2 + 11n + 5 \]

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Key Concepts

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

Binomial Multiplication
Binomial multiplication involves multiplying two expressions, each containing two terms. In our example, we have \((6n + 5)(n + 1)\). The terms in these expressions are separated by a plus or minus sign. When dealing with binomials, you systematically multiply each term from the first binomial by every term in the second binomial. This ensures every combination of terms is considered. This step-by-step approach helps in managing and simplifying the terms efficiently. Paying attention to every term helps avoid mistakes.
The general rule to follow is to distribute each term in the first binomial to each term in the second, ensuring all possible pairs of multiplication are covered.
Distributive Property
The distributive property is essential in binomial multiplication. It states that \(a(b+c) = ab + ac\). This rule allows you to multiply a single term by a binomial. When applied to our exercise, it looks like this:
We distribute each term in the first binomial, \((6n + 5)\), to every term in the second binomial, \((n + 1)\). Here鈥檚 how:
  • First, multiply \(6n\) with both \(n\) and \(1\):
    \(6n \times n = 6n^2\)
    \(6n \times 1 = 6n\)
  • Next, multiply \5\ with both \(n\) and \(1\):
    \(5 \times n = 5n\)
    \(5 \times 1 = 5\)
By carefully applying the distributive property, we transform and prepare the terms for the final step, which is combining.
Combining Like Terms
After multiplying the binomials and distributing the terms, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. Combining these terms simplifies the expression into a more manageable form.
In our example, after using the distributive property, we have: \(6n^2 + 6n + 5n + 5\). Here's the process:
  • Identify like terms. In this case, the terms with \ are \6n\ and \5n\.
  • Combine them by adding. So, \6n + 5n\ becomes \11n\.
Putting it all together, we get \(6n^2 + 11n + 5\). This is your simplified product, obtained by combining like terms correctly. This step ensures that you end up with the simplest possible polynomial.

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