Problem 64
Question
For exercises 39-82, simplify. $$ \frac{3 a+6}{a^{2}+4 a+4} \div \frac{a+4}{a+2} $$
Step-by-Step Solution
Verified Answer
\(\frac{3}{(a+2)(a+4)}\)
1Step 1: Rewrite the division as multiplication
Rewrite the division of fractions by multiplying with the reciprocal. \[\frac{3a+6}{a^2 + 4a + 4} \div \frac{a+4}{a+2} = \frac{3a+6}{a^2 + 4a + 4} \times \frac{a+2}{a+4}\]
2Step 2: Factor the numerators and denominators
Factorize the numerators and denominators to simplify easier. \[\frac{3(a+2)}{(a+2)^2} \times \frac{a+2}{a+4}\]
3Step 3: Cancel common factors
Cancel out the common factors present in both the numerator and the denominator. \[\frac{3 \cancel{(a+2)}}{(a+2) \cancel{(a+2)}} \times \frac{\cancel{(a+2)}}{a+4} = \frac{3}{a+2} \times \frac{1}{a+4}\]
4Step 4: Multiply the simplified fractions
Multiply the resulting fractions after cancelling out common factors. \[\frac{3}{a+2} \cdot \frac{1}{a+4} = \frac{3}{(a+2)(a+4)}\]
Key Concepts
Algebraic FractionsFactoring PolynomialsReciprocal MultiplicationCanceling Common FactorsMultiplying Fractions
Algebraic Fractions
Algebraic fractions are fractions where the numerator, the denominator, or both, contain algebraic expressions (like polynomials).
These are just like numerical fractions but involve variables. For example, in the expression \(\frac{3a+6}{a^2+4a+4}\), both the numerator and the denominator are polynomials.
Working with algebraic fractions often requires simplifying them, just like with numerical fractions.
Remember, you simplify by factoring expressions and canceling common factors.
These are just like numerical fractions but involve variables. For example, in the expression \(\frac{3a+6}{a^2+4a+4}\), both the numerator and the denominator are polynomials.
Working with algebraic fractions often requires simplifying them, just like with numerical fractions.
Remember, you simplify by factoring expressions and canceling common factors.
Factoring Polynomials
Factoring polynomials means breaking them down into simpler 'factor' polynomials that, when multiplied together, give the original polynomial.
For example, the polynomial \((a^2+4a+4)\) can be factored as \((a+2)(a+2)\) or \((a+2)^2\).
Factoring simplifies expressions and makes it easier to cancel out common factors later.
To factor polynomials, look for common terms, use special formulas like the difference of squares, or apply techniques such as grouping.
For example, the polynomial \((a^2+4a+4)\) can be factored as \((a+2)(a+2)\) or \((a+2)^2\).
Factoring simplifies expressions and makes it easier to cancel out common factors later.
To factor polynomials, look for common terms, use special formulas like the difference of squares, or apply techniques such as grouping.
Reciprocal Multiplication
When dividing fractions, you can instead multiply by the reciprocal (the 'flipped' version) of the second fraction.
In the given problem, we change \(\frac{3a+6}{a^2 + 4a + 4} \div \frac{a+4}{a+2}\) to \(\frac{3a+6}{a^2 + 4a + 4} \times \frac{a+2}{a+4}\).
This operation is called reciprocal multiplication.
Multiplying by the reciprocal simplifies the process of dividing fractions and is a crucial step to correctly solve such problems.
In the given problem, we change \(\frac{3a+6}{a^2 + 4a + 4} \div \frac{a+4}{a+2}\) to \(\frac{3a+6}{a^2 + 4a + 4} \times \frac{a+2}{a+4}\).
This operation is called reciprocal multiplication.
Multiplying by the reciprocal simplifies the process of dividing fractions and is a crucial step to correctly solve such problems.
Canceling Common Factors
Canceling common factors involves simplifying expressions by removing terms that appear in both the numerator and the denominator.
In the factored form, \(\frac{3(a+2)}{(a+2)^2} \times \frac{a+2}{a+4}\), the factor \((a+2)\) appears both in the numerator and denominator.
By canceling these common factors, we simplify the expression to \(\frac{3}{a+2} \times \frac{1}{a+4}\).
Always look for and eliminate common factors to simplify fractions before performing further operations.
In the factored form, \(\frac{3(a+2)}{(a+2)^2} \times \frac{a+2}{a+4}\), the factor \((a+2)\) appears both in the numerator and denominator.
By canceling these common factors, we simplify the expression to \(\frac{3}{a+2} \times \frac{1}{a+4}\).
Always look for and eliminate common factors to simplify fractions before performing further operations.
Multiplying Fractions
Multiplying fractions involves multiplying their numerators together and their denominators together.
After canceling the common factors in the simplified expression, we multiply: \(\frac{3}{a+2} \times \frac{1}{a+4} = \frac{3}{(a+2)(a+4)}\).
Write the final fraction in its simplest form.
Remember, always simplify fractions as much as possible before multiplying to make calculations easier and results clearer.
After canceling the common factors in the simplified expression, we multiply: \(\frac{3}{a+2} \times \frac{1}{a+4} = \frac{3}{(a+2)(a+4)}\).
Write the final fraction in its simplest form.
Remember, always simplify fractions as much as possible before multiplying to make calculations easier and results clearer.
Other exercises in this chapter
Problem 63
For exercises 39-82, simplify. $$ \frac{2 z+6}{z^{2}+3 z+2} \div \frac{z+3}{z+2} $$
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 65
For exercises \(65-68\), evaluate. $$ \sqrt{16} $$
View solution Problem 65
For exercises \(25-68\), evaluate or simplify. $$ \frac{3}{3+\frac{3}{3+x}} $$
View solution