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Chapter 7: Problem 7

Multiply. Leave each answer in factored form. $$\frac{7 x}{5} \cdot \frac{x-5}{2 x+1}$$

Short Answer

Expert verified
The factored form is \(\frac{7x(x-5)}{5(2x+1)}\).

Step by step solution

01

Write the Expression

Write the multiplication of the two fractions. \(\frac{7x}{5} \times \frac{x-5}{2x+1}\)
02

Multiply the Numerators

Multiply the numerators of the two fractions. The numerator is \(7x \times (x - 5)\).
03

Multiply the Denominators

Multiply the denominators of the two fractions. The denominator is \(5 \times (2x + 1)\).
04

Write the Result

Combine the products from Step 2 and Step 3 to write the final result. The final factored form is: \(\frac{7x(x-5)}{5(2x+1)}\)

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Key Concepts

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

Understanding Fractions
Fractions are made of two parts: the numerator and the denominator. The numerator is the number above the line, and the denominator is the number below the line.
For example, in the fraction \(\frac{3}{4}\), 3 is the numerator, and 4 is the denominator. Fractions represent parts of a whole.
Here鈥檚 a breakdown of why fractions are useful in algebra:
  • They help simplify complex expressions.
  • They are used to solve equations that involve parts of a whole.

When multiplying fractions, you multiply the numerators together and the denominators together, which is exactly what we are doing in our exercise.
Factored Form in Algebra
Factored form means writing an expression as a product of its factors. Factors are numbers or expressions that multiply together to make another number or expression.
In our exercise, the factored form makes it easier to see how the expressions interact through multiplication. For example, the factored form of the numerator in our exercise is 7x multiplied by (x - 5).
Benefits of using factored form:
  • It helps in simplifying expressions.
  • It makes solving equations easier.
  • It aids in finding roots and understanding the behavior of functions.

In step-by-step problems, maintaining factored form is crucial for clarity and accuracy.
Numerators and Denominators in Multiplication
In our exercise, the numerators are \({7x}\) and \(x - 5\), and the denominators are \(5\) and \(2x + 1\).
Here鈥檚 how we multiply them:
  • First, multiply the numerators: \(7x \times (x - 5)\).
  • Next, multiply the denominators: \(5 \times (2x + 1)\).

Finally, combine the products to get the fraction in factored form: \(\frac{7x(x-5)}{5(2x+1)}\). This process shows how the individual parts of the fractions transform when multiplied together, giving you a clearer view of the result.
Understanding how numerators and denominators work in multiplication helps you tackle a range of fraction problems with ease.

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Most popular questions from this chapter

Solve. $$\frac{\frac{1}{x}+1}{x}=\frac{\frac{1}{x}}{2}$$

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