Problem 63
Question
For exercises 39-82, simplify. $$ \frac{2 z+6}{z^{2}+3 z+2} \div \frac{z+3}{z+2} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \( \frac{2}{z+1} \).
1Step 1 - Factoring the Denominator
First, factor the quadratic expression in the denominator of the first fraction. The expression is \( z^2 + 3z + 2 \). It factors into \( (z+1)(z+2) \).
2Step 2 - Rewriting the Division
Rewrite the division as a multiplication by the reciprocal: \[ \frac{2z+6}{(z+1)(z+2)} \times \frac{z+2}{z+3} \].
3Step 3 - Simplifying the Numerator
Factor the numerator \( 2z + 6 \) of the first fraction to get \( 2(z+3) \). The expression now looks like this: \[ \frac{2(z+3)}{(z+1)(z+2)} \times \frac{z+2}{z+3} \].
4Step 4 - Canceling Common Factors
Cancel out the common factors \( (z+2) \) and \( (z+3) \) from the fractions: \[ \frac{2}{z+1} \].
5Step 5 - Final Simplification
The simplified form of the expression is \( \frac{2}{z+1} \).
Key Concepts
Factoring PolynomialsMultiplication by ReciprocalsCanceling Common FactorsSimplified Rational Expression
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler terms (factors) that, when multiplied together, give you the original polynomial. This is a crucial step in simplifying rational expressions.
In our original exercise, we start by factoring the quadratic polynomial in the denominator of the first fraction: \( z^2 + 3z + 2 \).
By finding two numbers that multiply to 2 and add to 3, we determine that these numbers are 1 and 2. Therefore, \( z^2 + 3z + 2 \) factors into \( (z+1)(z+2) \).
Here's a quick breakdown:
In our original exercise, we start by factoring the quadratic polynomial in the denominator of the first fraction: \( z^2 + 3z + 2 \).
By finding two numbers that multiply to 2 and add to 3, we determine that these numbers are 1 and 2. Therefore, \( z^2 + 3z + 2 \) factors into \( (z+1)(z+2) \).
Here's a quick breakdown:
- Find two numbers that multiply to the constant term (here: 2) and add to the coefficient of the linear term (here: 3).
- Rewrite the polynomial using these numbers as factors.
- Simplify the expression. For this exercise, it becomes \( (z+1)(z+2) \).
Multiplication by Reciprocals
When dividing rational expressions, it's often easier to rewrite the division as multiplication by the reciprocal of the divisor. This step can simplify the overall problem.
In our exercise, we have a division: \( \frac{2z+6}{(z+1)(z+2)} \div \frac{z+3}{z+2} \). To make this simpler, we convert it to multiplication:
\( \frac{2z+6}{(z+1)(z+2)} \times \frac{z+2}{z+3} \).
Here's how we do it:
In our exercise, we have a division: \( \frac{2z+6}{(z+1)(z+2)} \div \frac{z+3}{z+2} \). To make this simpler, we convert it to multiplication:
\( \frac{2z+6}{(z+1)(z+2)} \times \frac{z+2}{z+3} \).
Here's how we do it:
- Take the divisor (the second fraction) and find its reciprocal. The reciprocal of \( \frac{z+3}{z+2} \) is \( \frac{z+2}{z+3} \).
- Change the division sign to a multiplication sign and multiply the first fraction by this reciprocal.
Canceling Common Factors
Canceling common factors is an essential step in simplifying rational expressions. It involves removing any common factors from the numerator and denominator.
After rewriting the division as multiplication, our expression looks like this:
\( \frac{2(z+3)}{(z+1)(z+2)} \times \frac{z+2}{z+3} \).
Next, we look for common factors to cancel:
After rewriting the division as multiplication, our expression looks like this:
\( \frac{2(z+3)}{(z+1)(z+2)} \times \frac{z+2}{z+3} \).
Next, we look for common factors to cancel:
- We notice both the numerator of the second fraction and the denominator of the first fraction have \( z+2 \) as a common factor. Cancel out \( z+2 \).
- We also have \( z+3 \) as a common factor in the numerator of the first fraction and the denominator of the second fraction. Cancel out \( z+3 \).
Simplified Rational Expression
A simplified rational expression is one where the numerator and the denominator have no common factors other than 1. This is the goal of simplifying rational expressions.
In our exercise, after factoring, rewriting, and canceling common factors, we arrive at the simplified rational expression.
The original expression was: \( \frac{2z+6}{z^2+3z+2} \div \frac{z+3}{z+2} \).
Through steps of factoring the polynomials, converting division to multiplication by the reciprocal, and canceling common factors, we've simplified this to: \( \frac{2}{z+1} \).
This process streamlines the expression, making it easier to work with and understand. Remember, the fewer factors in the numerator and denominator, the simpler the expression.
In our exercise, after factoring, rewriting, and canceling common factors, we arrive at the simplified rational expression.
The original expression was: \( \frac{2z+6}{z^2+3z+2} \div \frac{z+3}{z+2} \).
Through steps of factoring the polynomials, converting division to multiplication by the reciprocal, and canceling common factors, we've simplified this to: \( \frac{2}{z+1} \).
This process streamlines the expression, making it easier to work with and understand. Remember, the fewer factors in the numerator and denominator, the simpler the expression.
Other exercises in this chapter
Problem 62
For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{c}{3}+\frac{d}{2}}{\frac{c}{2}+\frac{d}{7}} $$
View solution Problem 62
For exercises 1-66, simplify. $$ \frac{2 c^{3}-2 c^{2}-4 c}{4 c^{3}-8 c^{2}-12 c} $$
View solution Problem 64
For exercises 61-64, the completed problem has one mistake. (a) Describe the mistake in words or copy down the whole problem and highlight or circle the mistake
View solution Problem 64
For exercises 39-82, simplify. $$ \frac{3 a+6}{a^{2}+4 a+4} \div \frac{a+4}{a+2} $$
View solution