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Chapter 7: Problem 11

Multiply. Leave each answer in factored form. $$\frac{2 x+3}{4} \cdot \frac{x+1}{x-5}$$

Short Answer

Expert verified
\( \frac{(2x + 3)(x + 1)}{4(x - 5)} \)

Step by step solution

01

- Identify the fractions to multiply

The given fractions are \(\frac{2x+3}{4}\) and \(\frac{x+1}{x-5}\). We need to multiply these fractions.
02

- Apply the multiplication rule for fractions

To multiply fractions, multiply the numerators together and multiply the denominators together. The rule is \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\).
03

- Multiply the numerators

Multiply the numerators \(2x + 3\) and \(x + 1\): \( (2x + 3) \times (x + 1)\).
04

- Multiply the denominators

Multiply the denominators \4\ and \(x - 5\): \( 4 \times (x - 5) \).
05

- Write the product of the fractions in factored form

Combine the results from steps 3 and 4. The product will be: \(\frac{(2x + 3)(x + 1)}{4(x - 5)}\).

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Key Concepts

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

factored form
When multiplying fractions, it's often best to leave the answer in factored form. This means we do not need to simplify or expand the expressions. Factored form helps in easily identifying common factors between numerators and denominators. For example, with our exercise, after multiplying the fractions, the result is \(\frac{(2x + 3)(x + 1)}{4(x - 5)}\).
By keeping the answer in factored form, we see all components clearly without expanding them.
numerators and denominators
Fractions have two main parts: numerators on the top and denominators on the bottom. In our exercise, the first fraction is \(\frac{2x + 3}{4}\). Here, '2x + 3' is the numerator and '4' is the denominator.
The second fraction is \(\frac{x+1}{x-5}\), with 'x + 1' as the numerator and 'x - 5' as the denominator.
Understanding these parts is crucial because when we multiply fractions, we deal with numerators and denominators separately.
multiplication rule for fractions
The multiplication rule for fractions tells us to multiply the numerators together and the denominators together.
Mathematically, if we have two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), their product is \(\frac{ac}{bd}\).
In our exercise, the numerators are \(2x + 3\) and \(x + 1\). By multiplying them, we get \((2x + 3) \times (x + 1)\).
The denominators are '4' and \(x - 5\). By multiplying these, we get \(4 \times (x - 5)\).
Putting it all together, our result in factored form is \(\frac{(2x + 3)(x + 1)}{4(x - 5)}\).

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