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Chapter 7: Problem 79

For exercises 39-82, simplify. $$ \frac{36 a^{2}+12 a+1}{18 a^{2}+15 a+2} \div \frac{6 a^{2}-17 a-3}{3 a^{2}-16 a-12} $$

Short Answer

Expert verified
\( \frac{(6a + 1)(a - 6)}{(3a + 2)(3a + 1)} \)

Step by step solution

01

- Rewrite the Division as Multiplication

Rewrite the division of fractions as multiplication by the reciprocal of the second fraction. \[ \frac{36 a^{2}+12 a+1}{18 a^{2}+15 a+2} \times \frac{3 a^{2}-16 a-12}{6 a^{2}-17 a-3} \]
02

- Factor the Numerators and Denominators

Factor each of the polynomials in both fractions.For the first numerator: \[36a^2 + 12a + 1 = (6a + 1)^2\]For the first denominator: \[18a^2 + 15a + 2 = (3a + 2)(6a + 1)\]For the second numerator: \[3a^2 - 16a - 12 = (3a + 2)(a - 6)\]For the second denominator: \[6a^2 - 17a - 3 = (3a + 1)(2a - 3)\]
03

- Substitute the Factored Forms into the Expression

Replace the polynomials with their factored forms in the expression: \[ \frac{(6a + 1)^2}{(3a + 2)(6a + 1)} \times \frac{(3a + 2)(a - 6)}{(3a + 1)(2a - 3)} \]
04

- Cancel Common Factors

Cancel any common factors in the numerators and denominators. The common factors \((6a + 1)\) and \((3a + 2)\) can be canceled: \[ \frac{6a + 1}{3a + 2} \times \frac{a - 6}{3a + 1} \]
05

- Simplify the Expression

After canceling, multiply the remaining fractions together: \[ \frac{(6a + 1)(a - 6)}{(3a + 2)(3a + 1)} \]

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Key Concepts

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

Factoring Polynomials
Factoring polynomials means breaking down a polynomial into simpler polynomials whose product is the original polynomial. This makes it easier to work with. In our exercise, we factor each polynomial expression in the numerators and denominators.
For example, we start with these polynomials:
  • For the numerator:
    • \[36a^2 + 12a + 1\text{ turns into } (6a + 1)^2\text{.} \]
  • For the denominator:
    • \[18a^2 + 15a + 2\text{ turns into } (3a + 2)(6a + 1)\text{.} \]
By factoring these polynomials, we can easily cancel out common factors later on.

Factoring is a critical step in simplifying rational expressions and solving polynomial equations.
Rational Expressions
A rational expression is a fraction where the numerator and the denominator are polynomials. In algebra, rational expressions need to be simplified by factoring and canceling common terms.
In our exercise, the original expression is a division of two rational expressions:
\[\frac{36 a^{2}+12 a+1}{18 a^{2}+15 a+2} \div \frac{6 a^{2}-17 a-3}{3 a^{2}-16 a-12}\]
We rewrite the division as multiplication by the reciprocal:
  • \[\frac{36 a^{2}+12 a+1}{18 a^{2}+15 a+2} \times \frac{3 a^{2}-16 a-12}{6 a^{2}-17 a-3}\]
Once we convert the division to multiplication, we can factor each polynomial and simplify the expression by canceling common factors.

This approach makes the expression much simpler to handle and solve.
Simplifying Algebraic Fractions
Simplifying algebraic fractions involves reducing them to their simplest form by factoring and canceling common factors.
In our exercise, after factoring, we substitute the factored forms back into the expression:
\[\frac{(6a + 1)^2}{(3a + 2)(6a + 1)} \times \frac{(3a + 2)(a - 6)}{(3a + 1)(2a - 3)}\]
We then cancel common factors
  • \[(6a + 1)\text{ and }(3a + 2)\]
This leaves us with:
\[\frac{6a + 1}{3a + 2} \times \frac{a - 6}{3a + 1}\]
Finally, we multiply the remaining expressions:
\[\frac{(6a + 1)(a - 6)}{(3a + 2)(3a + 1)}\]

This is the fully simplified form of the original rational expression. By simplifying algebraic fractions, we make it easier to understand and solve mathematical problems.

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Most popular questions from this chapter

When the top of a cone is removed, the formula for the volume of the remaining cone (the frustrum) is \(V=\frac{1}{3} \pi\left(R^{2}+R r+r^{2}\right) h\), where \(r\) is the radius of the circle at the top of the frustrum and \(R\) is the radius of the circle at the bottom of the frustrum. In 1856, an American army officer, Henry Hopkins Sibley, invented and received a patent for the design of a conical tent that could sleep 12 soldiers. (The apex is the diameter of the top of the frustrum.) Find the volume of the tent in cubic feet. Use \(\pi \approx 3.14\). Round to the nearest whole number. Be it known that I, H.H. Sibley, United States Army, have invented a new and improved Conical Tent ... the tent is in shape the frustrum of a cone; the base 18 feet; the height 12 feet; the apex 1 foot 6 inches [1.5 ft]. (Source: patimg1.uspto.gov)

Author Colin Tudge writes in his book The Variety of Life: "For some have argued that works of art should be self-contained and need no extraneous information to be appreciated: no biography, no history, no referents of any kind." Explain the meaning of extraneous in this statement. (Source: Word of the Day, October 27, 2000, Merriam-Webster Online)

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 direct variation or an inverse variation? $$ R \text { and } P \text { are constant; the relationship of } U \text { and } F \text {. } $$

For exercises 49-52, the formula \(C=\frac{P_{m} P_{i}}{T F}\) describes the cost of insurance, \(C\). Is the relationship of the given variables a direct variation or an inverse variation? $$ P_{m}, T \text {, and } F \text { are constant; the relationship of } C \text { and } P_{i} $$

$$ \text { Solve: } 800=5 k $$
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