Problem 46 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{3 b^{2}+10 b+3}{3 b^{2}+7 b+2} $$

Step-by-Step Solution

Answer

Question: Reduce the rational expression \(\frac{3b^2 + 10b + 3}{3b^2 + 7b + 2}\) to its lowest terms. Answer: The reduced rational expression is \(\frac{b+3}{b+2}\).

Step 1: Factor the numerator and the denominator

We will start by factoring the numerator and the denominator of the given expression. Numerator: \(3b^2 + 10b + 3\) Denominator: \(3b^2 + 7b + 2\) First, let's factor the numerator: $$ 3b^2 + 10b + 3 = (3b + 1)(b + 3) $$ Now, let's factor the denominator: $$ 3b^2 + 7b + 2 = (3b + 1)(b + 2) $$

Step 2: Cancel common factors

Now that we have factored the numerator and the denominator, we can cancel out the common factors \((3b+1)\): $$ \frac{(3b+1)(b+3)}{(3b+1)(b+2)} $$ Since \((3b+1)\) appears in both the numerator and the denominator, we can cancel it out: $$ \frac{b+3}{b+2} $$

Step 3: Write the simplified expression

After canceling the common factors, our simplified, reduced rational expression is: $$ \frac{b+3}{b+2} $$

Key Concepts

Factoring Polynomials

Factoring polynomials is a vital skill when dealing with rational expressions. It involves breaking down a complex polynomial into simpler "factors" that, when multiplied back together, give you the original polynomial. Let's look at this in action with the given problem's numerator and denominator.

The numerator is a quadratic polynomial: \(3b^2 + 10b + 3\). To factor it, we looked for two numbers that multiply to give the product of the first and last coefficients (in this case, \(3 \times 3 = 9\)) and add up to the middle coefficient (here, 10). We found that the numbers 9 and 1 fulfill this condition. Using these, we factored the quadratic as \((3b + 1)(b + 3)\).

Similarly, the denominator \(3b^2 + 7b + 2\) follows the same method. Our goal was to find two numbers multiplying to \(6\) (product of 3 and 2) that add to 7. These numbers turn out to be 6 and 1. Thus, the factorization became \((3b + 1)(b + 2)\).
Factoring helps simplify rational expressions by making common terms easy to identify and cancel.

Simplifying Fractions

Simplifying fractions involves making a fraction as simple as possible. The initial step involves reducing the fraction by canceling out any common factors between the numerator and denominator.

In the given problem, once we factored both \(3b^2 + 10b + 3\) and \(3b^2 + 7b + 2\) into \((3b + 1)(b + 3)\) and \((3b + 1)(b + 2)\) respectively, we identified \((3b + 1)\) as a common factor that appears in both the numerator and the denominator.

By "canceling" this common factor, we effectively divide both the top and bottom by this factor, thereby simplifying the fraction. After cancellation, the expression was reduced to \(\frac{b+3}{b+2}\).
This process not only simplifies arithmetic but also clarifies what remains in the simplified expression.

Lowest Terms

Writing a rational expression in "lowest terms" means simplifying it to the point where no common factors exist between the numerator and the denominator. This is much like reducing a regular fraction, such as simplifying \(\frac{4}{8}\) to \(\frac{1}{2}\).

In our example, the original expression \(\frac{3b^2 + 10b + 3}{3b^2 + 7b + 2}\) was in more complex terms initially. After factoring and canceling, we achieved \(\frac{b+3}{b+2}\), which has no common factors other than 1. This demonstrates the expression being "in lowest terms."

By reducing to lowest terms, we ensure the expression is as simple as possible, making further algebraic operations more manageable and avoiding unnecessary complexity. Furthermore, any further calculations or evaluations based on this fraction are much simpler when it's in its simplest form.