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Chapter 5: Problem 36

Multiply. \(\left(x+\frac{3}{5}\right)\left(x-\frac{2}{5}\right)\)

Short Answer

Expert verified
The product is \(x^2 + \frac{1}{5}x - \frac{6}{25}\).

Step by step solution

01

Identify the Expression Form

The expression to be multiplied is \((x + \frac{3}{5})(x - \frac{2}{5})\). This is a product of two binomials.
02

Apply the FOIL Method

FOIL stands for First, Outside, Inside, Last, which are the terms we multiply together when we have two binomials. For \((x + \frac{3}{5})(x - \frac{2}{5})\), apply the method as follows:- **First**: Multiply the first terms: \(x \cdot x = x^2\)- **Outside**: Multiply the outer terms: \(x \cdot -\frac{2}{5} = -\frac{2}{5}x\)- **Inside**: Multiply the inner terms: \(\frac{3}{5} \cdot x = \frac{3}{5}x\)- **Last**: Multiply the last terms: \(\frac{3}{5} \cdot -\frac{2}{5} = -\frac{6}{25}\)
03

Combine Like Terms

After applying FOIL, we have: \[x^2 - \frac{2}{5}x + \frac{3}{5}x - \frac{6}{25}\].Combine the like terms (the \(x\) terms): \(-\frac{2}{5}x + \frac{3}{5}x = \frac{1}{5}x\).Thus, the expression becomes: \[x^2 + \frac{1}{5}x - \frac{6}{25}\].

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Key Concepts

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

Multiplying Binomials
When multiplying binomials, such as in the expression \[(x + \frac{3}{5})(x - \frac{2}{5}),\]there鈥檚 a handy tool we often use called the FOIL method. Each letter in FOIL stands for a different pair of terms that need to be multiplied:
  • First: This refers to the first terms in each binomial (in our case, the \(x\) terms), giving \(x \cdot x = x^2\).
  • Outside: This refers to the outer terms of the entire expression, which is \(x\) from the first binomial and \(-\frac{2}{5}\) from the second, resulting in \(x \cdot -\frac{2}{5} = -\frac{2}{5}x\).
  • Inside: Here, you multiply the inner terms, \(\frac{3}{5}\) and \(x\), to get \(\frac{3}{5} \cdot x = \frac{3}{5}x\).
  • Last: Finally, you multiply the last terms in each binomial: \(\frac{3}{5} \cdot -\frac{2}{5} = -\frac{6}{25}\).
The result, before combining like terms, is:\[x^2 - \frac{2}{5}x + \frac{3}{5}x - \frac{6}{25}.\]
Combining Like Terms
After using the FOIL method, we end up with several terms, some of which can be simplified. The like terms in our expression \[x^2 - \frac{2}{5}x + \frac{3}{5}x - \frac{6}{25}\]are the terms that contain the variable \(x\). In this case, these are \(-\frac{2}{5}x\) and \(\frac{3}{5}x\).

Combining like terms involves adding or subtracting the coefficients of these terms. Here, we calculate:
  • \(-\frac{2}{5}x + \frac{3}{5}x = \frac{1}{5}x\)
This simplifies the entire expression to:\[x^2 + \frac{1}{5}x - \frac{6}{25}.\]By simplifying, you make the polynomial neater and often easier to analyze or use in further calculations.
Polynomial Multiplication
Polynomial multiplication expands beyond the simple multiplication of numbers. It's about multiplying each term in one polynomial by each term in another.
  • In the case of binomials, like \((x + \frac{3}{5})\) and \((x - \frac{2}{5})\), each term in the first binomial is multiplied by each term in the second one.
This method ensures that every possible product between both polynomials is accounted for.

In general, multiplying polynomials involves repeating this careful process, making sure to line up like terms as you go. This keeps your work organized and ensures you don't miss any necessary calculations. As with the FOIL method, practice helps this process become smooth and intuitive.

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Most popular questions from this chapter

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