Problem 79 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1 - Rewrite the Division as Multiplication

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

Step 2 - 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: \[6a^2 - 17a - 3 = (3a + 1)(2a - 3)\]

Step 3 - Substitute the Factored Forms into the Expression

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

Step 4 - Cancel Common Factors

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

Step 5 - Simplify the Expression

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

Key Concepts

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.