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Chapter 7: Problem 48

For exercises 1-66, simplify. $$ \frac{y^{2}+11 y+30}{y^{2}-2 y-48} $$

Short Answer

Expert verified
The simplified expression is \( \frac{y + 5}{y - 8} \).

Step by step solution

01

Factor the Numerator

The numerator is a quadratic expression: \( y^2 + 11y + 30 \). Find two numbers that multiply to 30 and add to 11. The numbers are 5 and 6. Thus, we can factor the numerator as: \( (y + 5)(y + 6) \).
02

Factor the Denominator

The denominator is also a quadratic expression:\( y^2 - 2y - 48 \). Find two numbers that multiply to -48 and add to -2. The numbers are -8 and 6. Thus, we can factor the denominator as: \( (y - 8)(y + 6) \).
03

Simplify the Expression

Now that both the numerator and the denominator are factored, the expression becomes: \( \frac{(y + 5)(y + 6)}{(y - 8)(y + 6)} \). Cancel the common factor of \( (y + 6) \) in the numerator and the denominator. The expression simplifies to: \( \frac{y + 5}{y - 8} \).

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Key Concepts

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

Factoring Quadratics
Quadratic equations can often be simplified by factoring them. This means breaking down the equation into simpler expressions that, when multiplied together, give you the original quadratic expression. For example, the quadratic expressions in the numerator and denominator of our exercise are:
Numerator: \( y^2 + 11y + 30 \)
Denominator: \( y^2 - 2y - 48 \)
These can be factored by finding two numbers that multiply to the constant term and add up to the coefficient of the linear term:
  • For the numerator: 5 * 6 = 30 and 5 + 6 = 11, giving us \( (y + 5)(y + 6) \)
  • For the denominator: -8 * 6 = -48 and -8 + 6 = -2, giving us \( (y - 8)(y + 6) \)
Algebraic Simplification
After factoring, the next step is algebraic simplification. The goal here is to see if there are common factors in both the numerator and the denominator that can be canceled out. In our case, the factored form of the expression is:
\( \frac{(y + 5)(y + 6)}{(y - 8)(y + 6)} \)
We notice that \( (y + 6) \) is a common factor in both the numerator and the denominator. By canceling out this common factor, we simplify the expression to:
\( \frac{y + 5}{y - 8} \)
This process is critical for simplifying complex expressions and making them more manageable.
Numerator and Denominator
The numerator is the top part of a fraction, and the denominator is the bottom part. When simplifying rational expressions, it's crucial to understand these components. In our example, the original numerator and denominator were quadratic expressions:
Numerator: \( y^2 + 11y + 30 \)
Denominator: \( y^2 - 2y - 48 \)
After factoring and simplification, the numerator becomes \( y + 5 \) and the denominator becomes \( y - 8 \). Understanding the roles of the numerator and denominator helps you see how factors can be canceled, which is key to simplifying rational expressions effectively.

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Most popular questions from this chapter

In radiography, a grid reduces the effect of X-ray scattering. The relationship of the interspace distance on the grid, \(x\), and the grid ratio, \(y\), is an inverse variation. When the interspace distance on a grid is 300 micrometers, the grid ratio is 8 . Write an equation that represents this variation. Include the units.

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=10\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=5\).

For exercises 31-40, (a) solve. (b) check. $$ \frac{3}{w-3}+\frac{4}{w}=\frac{w}{w-3} $$

The relationship of \(x\) and \(y\) is a direct variation. When \(x=1, y=5\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this direct variation. c. Find \(y\) when \(x=2\). d. Use slope-intercept graphing to graph this equation. e. Use the graph to find \(y\) when \(x=3\).

For exercises 1-10, (a) solve. (b) check. $$ \frac{2}{9} x+\frac{5}{3}=\frac{5}{9} x+\frac{7}{3} $$
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