Problem 47
Question
For exercises 1-66, simplify. $$ \frac{y^{2}+11 y+28}{y^{2}-2 y-63} $$
Step-by-Step Solution
Verified Answer
\( \frac{y + 4}{y - 9} \)
1Step 1 - Factor the numerator
The numerator is a quadratic expression: \(y^2 + 11y + 28 \). Look for two numbers that multiply to 28 and add up to 11. Those numbers are 4 and 7. Therefore, we can factor the numerator as: \( (y + 4)(y + 7) \)
2Step 2 - Factor the denominator
The denominator is another quadratic expression: \(y^2 - 2y - 63\). Look for two numbers that multiply to -63 and add up to -2. Those numbers are -9 and 7. Thus, we can factor the denominator as: \( (y - 9)(y + 7) \)
3Step 3 - Simplify the expression
Now that we have factored both the numerator and denominator, we can rewrite the original fraction: \( \frac{(y + 4)(y + 7)}{(y - 9)(y + 7)} \). Both the numerator and the denominator have a common factor of \( (y + 7) \). Cancel out the common factor: \( \frac{(y + 4)}{(y - 9)} \)
Key Concepts
Factoring QuadraticsCanceling Common FactorsRational Expressions
Factoring Quadratics
Factoring quadratic expressions is a crucial skill in simplifying rational expressions. Quadratics are polynomial expressions of the form \(ax^2 + bx + c\). To factor them, we need to find two numbers that multiply to give the constant term \(c\), and add up to the coefficient of the linear term \(b\).
For instance, in the given problem, the numerator is \(y^2 + 11y + 28\). We need numbers that multiply to 28 and add up to 11. These numbers are 4 and 7. Hence, we can factor the numerator as follows:
\(y^2 + 11y + 28 = (y + 4)(y + 7)\)
Similarly, for the denominator \(y^2 - 2y - 63\), we need numbers that multiply to -63 and add up to -2. The numbers -9 and 7 fit the criteria. Thus, the denominator factors into:
\(y^2 - 2y - 63 = (y - 9)(y + 7)\)
Factoring quadratics helps decompose complex expressions into simpler factors, making subsequent steps easier.
For instance, in the given problem, the numerator is \(y^2 + 11y + 28\). We need numbers that multiply to 28 and add up to 11. These numbers are 4 and 7. Hence, we can factor the numerator as follows:
\(y^2 + 11y + 28 = (y + 4)(y + 7)\)
Similarly, for the denominator \(y^2 - 2y - 63\), we need numbers that multiply to -63 and add up to -2. The numbers -9 and 7 fit the criteria. Thus, the denominator factors into:
\(y^2 - 2y - 63 = (y - 9)(y + 7)\)
Factoring quadratics helps decompose complex expressions into simpler factors, making subsequent steps easier.
Canceling Common Factors
Once both the numerator and denominator are factored, the next step in simplifying rational expressions is to cancel out common factors. Common factors are identical expressions present in both the numerator and denominator.
In our example, we factored the rational expression as follows:
\(\frac{(y + 4)(y + 7)}{(y - 9)(y + 7)}\)
Notice that \(y + 7\) is present in both the numerator and the denominator. We can cancel out \(y + 7\) from both parts of the fraction, leaving us with:
\(\frac{(y + 4)}{(y - 9)}\)
By canceling the common factors, we simplify the expression to its simplest form. It is essential to remember that canceling is only valid when the common factor does not equal zero, as division by zero is undefined.
In our example, we factored the rational expression as follows:
\(\frac{(y + 4)(y + 7)}{(y - 9)(y + 7)}\)
Notice that \(y + 7\) is present in both the numerator and the denominator. We can cancel out \(y + 7\) from both parts of the fraction, leaving us with:
\(\frac{(y + 4)}{(y - 9)}\)
By canceling the common factors, we simplify the expression to its simplest form. It is essential to remember that canceling is only valid when the common factor does not equal zero, as division by zero is undefined.
Rational Expressions
Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying these expressions involves factoring and canceling common factors, as demonstrated in the previous sections.
A rational expression retains its value when simplified, provided the variable values do not make the denominator zero.
For example, with the expression:
\(\frac{y^2 + 11y + 28}{y^2 - 2y - 63}\)
Factored to:
\(\frac{(y + 4)(y + 7)}{(y - 9)(y + 7)}\)
And then simplified to:
\(\frac{(y + 4)}{(y - 9)}\)
The simplified form is valid as long as \(y eq -7 \text{ or } 9\), because these values would make the original and simplified denominators zero, leading to an undefined fraction.
Understanding rational expressions and their simplification helps solve many algebraic problems more efficiently.
A rational expression retains its value when simplified, provided the variable values do not make the denominator zero.
For example, with the expression:
\(\frac{y^2 + 11y + 28}{y^2 - 2y - 63}\)
Factored to:
\(\frac{(y + 4)(y + 7)}{(y - 9)(y + 7)}\)
And then simplified to:
\(\frac{(y + 4)}{(y - 9)}\)
The simplified form is valid as long as \(y eq -7 \text{ or } 9\), because these values would make the original and simplified denominators zero, leading to an undefined fraction.
Understanding rational expressions and their simplification helps solve many algebraic problems more efficiently.
Other exercises in this chapter
Problem 47
For exercises \(5-48\), simplify. $$ \frac{2 v^{2}}{2 v^{2}+5 v-12}+\frac{13 v}{2 v^{2}+5 v-12}-\frac{24}{2 v^{2}+5 v-12} $$
View solution Problem 47
For exercises 39-82, simplify. $$ \frac{8 a b}{21 c^{2}} \div \frac{2 a^{2}}{3 c} $$
View solution Problem 48
For exercises \(45-48\), the formula \(R=\frac{U F}{P}\) describes the glomular filtration rate by a kidney \(R\). Is the relationship of the given variables a
View solution Problem 48
For exercises 43-58, (a) solve. (b) check. $$ \frac{4}{a+6}=\frac{9}{a-4} $$
View solution