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Chapter 5: Problem 246

Find each product. $$ (7 a+b)(a-7 b) $$

Short Answer

Expert verified
= 7a^2 - 48ab - 7b^2

Step by step solution

01

- Apply the Distributive Property

Use the distributive property to expand the expression \[\begin{equation} onumber \begin{aligned} (7a + b)(a - 7b) &= (7a) \times a + (7a) \times (-7b) + (b) \times a + (b) \times (-7b). onumber \ &= 7a^2 - 49ab + ab - 7b^2. onumber \ \text{ Combine like terms:} &= 7a^2 - 48ab - 7b^2. onumber \ \text{So, the product is} &\boldsymbol{7a^2 - 48ab - 7b^2.} onumber \ \end{aligned} \end{equation}\]

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Key Concepts

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

Understanding the Distributive Property
The distributive property is a fundamental concept in algebra that allows us to simplify expressions and solve equations more easily. It states that multiplying a sum by a number equals multiplying each addend individually by the number and then adding the products.
In mathematical terms, it looks like this: \(a(b + c) = ab + ac\).
In the given exercise, we applied the distributive property to expand the expression \((7a + b)(a - 7b)\).
We multiplied each term in the first binomial by each term in the second binomial. Breaking it down:
  • Multiply \(7a\) by \(a\) \(\Rightarrow 7a^2\)
  • Multiply \(7a\) by \(-7b\) \(\Rightarrow -49ab\)
  • Multiply \(b\) by \(a\) \(\Rightarrow ab\)
  • Multiply \(b\) by \(-7b\) \(\Rightarrow -7b^2\)
These individual products are then added together to get: \(7a^2 - 49ab + ab - 7b^2\).
Combining Like Terms
Combining like terms simplifies an algebraic expression by consolidating terms that have the same variable raised to the same power. This step is crucial to make expressions easier to understand and solve.
In our expression \(7a^2 - 49ab + ab - 7b^2\), we can see there are like terms:\(-49ab\) and \(ab\). Both have the same variable combination \(ab\).
To combine them, we simply add their coefficients:
  • -49 and 1 (implicitly, because \(ab\) is the same as \(1ab\)) \(\Rightarrow -49ab + ab\) equals \(-48ab\)
After combining like terms, our simplified expression becomes \(7a^2 - 48ab - 7b^2\). Combining like terms helps to make complex algebraic expressions more manageable.
Working with Binomials
A binomial is an algebraic expression that contains exactly two terms separated by a plus or minus sign. Examples include expressions like \(a + b\) or \(7a - 4b\). Understanding how to work with binomials is a significant part of algebra.
When multiplying binomials, such as in the exercise \((7a + b)(a - 7b)\), we use the distributive property to find each product and then combine like terms to simplify.
This process involves a few straightforward steps:
  • Distribute each term in the first binomial across the second binomial.
  • Add or subtract like terms.
  • Simplify the resulting expression.
These steps ensure accurate results when dealing with polynomial expressions. By mastering binomials, you're building a strong foundation for more complex algebraic concepts.

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