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Magazine Chapter 8: Problem 57
Multiply or divide as indicated. $$\frac{4 t^{2}+12 t+36}{4 t-2} \cdot \frac{t^{2}-9}{t^{3}-27}$$
Short Answer
Expert verified
The simplified expression is: \(\frac{2(2t + 6)}{(2t - 1)(t - 3)}\).
Step by step solution
01
Factor the expressions
First, we will factor each of the expressions in the numerators and denominators.
Factor the first expression \(\frac{4 t^{2}+12 t+36}{4 t-2}\):
Numerator: \(4t^2 + 12t + 36\)\(= 4(t^2 + 3t + 9)\)
Denominator: \(4t - 2\)\(= 2(2t - 1)\)
Factor the second expression \(\frac{t^{2}-9}{t^{3}-27}\):
Numerator: \(t^2 - 9\)\(= (t - 3)(t + 3)\)
Denominator: \(t^3 - 27\)\(= (t - 3)(t^2 + 3t + 9)\)
Now the factored expressions are:
\(\frac{4(t^2 + 3t + 9)}{2(2t - 1)}\) \(\cdot\) \(\frac{(t - 3)(t + 3)}{(t - 3)(t^2 + 3t + 9)}\)
02
Cancel common factors
Next, we will cancel out the common factors in the numerators and denominators.
The common factors between the numerators and denominators are \((t-3)\) and \((t^2 + 3t + 9)\).
So, after canceling the common factors, the expressions become:
\(\frac{4}{2(2t - 1)}\) \(\cdot\) \(\frac{(t + 3)}{(t - 3)}\)
03
Perform the multiplication
Now that we have simplified the expressions as much as possible, we will multiply the remaining factors.
\(\frac{4}{2(2t - 1)}\) \(\cdot\) \(\frac{(t + 3)}{(t - 3)} = \frac{4 \cdot (t + 3)}{2(2t - 1)(t - 3)}\)
04
Simplify the result
Finally, we will simplify the result.
\(\frac{4 \cdot (t + 3)}{2(2t - 1)(t - 3)} = \frac{2(2t + 6)}{(2t - 1)(t - 3)} = \frac{2(2t + 6)}{(2t - 1)(t - 3)}\)
The final result is: \(\frac{2(2t + 6)}{(2t - 1)(t - 3)}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Polynomials
Factoring is a crucial concept when dealing with algebraic fractions, especially when simplifying or dividing expressions. It involves writing a polynomial as a product of its factors. This is much like breaking down a number into its prime components. Let's explore the key steps in factorization:
- **Identify a Common Factor:** Begin by looking for the greatest common factor (GCF) among the terms of the polynomial. This step was particularly essential for our first polynomial, where 4 was the GCF for the numerator: \(4t^2 + 12t + 36\).
- **Recognize Special Patterns:** Be alert for special product patterns, such as the difference of squares, \(a^2 - b^2 = (a-b)(a+b)\), which was used to factor \(t^2 - 9\) in the exercise.
- **Factor Completely:** If the polynomial is a cubic or higher and recognizable as a difference or sum of cubes, factor accordingly. In this case, \(t^3-27\) was recognized as a difference of cubes: \((t-3)(t^2+3t+9)\).
Simplifying Fractions
Simplifying algebraic fractions means reducing them to their simplest form. This process involves both factoring and cancellation.
- **Factor Both Numerator and Denominator:** After factorization, you can see common factors more easily. For instance, after factoring our original fractions, we had several opportunities to simplify by cancellation.
- **Cancel Common Factors:** If the same factor exists in both the numerator and denominator, they "cancel out" because they divide to 1, simplifying the expression. In the exercise, \((t-3)\) and \((t^2+3t+9)\) were common between numerators and denominators, allowing us to cancel them out, simplifying the expression significantly.
- **Re-evaluate Remaining Terms:** Always double-check the remaining terms for further simplification opportunities. Simplifying makes expressions manageable and easier to work with in subsequent operations.
Polynomial Division
Polynomial division refers to dividing a polynomial by another. This typically involves simplifying expressions and is often followed by multiplication to consolidate results.
- **Multiply Remaining Expressions:** Once common factors are canceled, what's left needs to be multiplied between numerators and denominators. In the exercise, after canceling, we multiplied \(4\) by \((t + 3)\) in the numerator, and factored products in the denominator.
- **Verify and Simplify:** Post multiplication, re-check for the possibility of further simplification. The exercise led us to a clean expression, \(\frac{4(2t+6)}{2(2t-1)(t-3)}\). Further simplification yielded \(\frac{2(2t+6)}{(2t-1)(t-3)}\), confirming that every term was reviewed for potential reduction.
- **Arrange Final Expression:** Write the final result in a succinct and clear way, ensuring it highlights the most manageable form of the original complex algebraic fraction.
