Problem 47 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{3 x^{2}+7 x+2}{3 x^{2}+13 x+4} $$

Step-by-Step Solution

Answer

\(\frac{x + 2}{x + 4}\) is the simplified expression.

Step 1: Factor the Numerator

We need to factor the numerator, which is \(3x^2 + 7x + 2\). For quadratic expressions of the form \(ax^2 + bx + c\), find two numbers that multiply to \(ac\) (which is \(3 \times 2 = 6\)) and add to \(b\) (which is \(7\)). These two numbers are \(6\) and \(1\). So, we rewrite the middle term as \(6x + x\). The expression becomes \(3x^2 + 6x + x + 2\). Next, factor by grouping: \((3x^2 + 6x) + (x + 2)\). Factor out \(3x\) from the first group and \(1\) from the second group, giving \(3x(x + 2) + 1(x + 2)\). This factors further to \((3x + 1)(x + 2)\).

Step 2: Factor the Denominator

Now we factor the denominator, which is \(3x^2 + 13x + 4\). Again look for two numbers that multiply to \(3 \times 4 = 12\) and add to \(13\). These numbers are \(12\) and \(1\). Split the middle term to rewrite: \(3x^2 + 12x + x + 4\). Group and factor: \((3x^2 + 12x) + (x + 4)\). Factor out \(3x\) from the first group and \(1\) from the second group, resulting in \(3x(x + 4) + 1(x + 4)\). Finally, factor to \((3x + 1)(x + 4)\).

Step 3: Simplify the Expression

The expression is now \(\frac{(3x + 1)(x + 2)}{(3x + 1)(x + 4)}\). Since \((3x + 1)\) is common to both the numerator and the denominator, they cancel each other out. Simplifying, we are left with \(\frac{x + 2}{x + 4}\).

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential algebraic skill that allows us to simplify expressions, especially when dealing with quadratic equations of the form \(ax^2 + bx + c\). To factor such an expression, we look for two numbers that multiply to the product \(ac\) and add up to \(b\). This method, known as factoring by grouping, is quite handy and straightforward once mastered.
Let's take the expression \(3x^2 + 7x + 2\) as an example. First, we multiply the coefficient of \(x^2\) (which is 3) by the constant term (which is 2), giving us 6. Now, we need two numbers that multiply to 6 and add to 7, our \(b\) term. These numbers are 6 and 1.
With these numbers in hand, we rewrite the middle term, splitting it into two separate terms: \(6x + x\). So, our expression \(3x^2 + 7x + 2\) becomes \(3x^2 + 6x + x + 2\). We can then group and factor: From \(3x^2 + 6x\), factor out \(3x\), and from \(x + 2\), factor out 1. After factoring, the expression simplifies to \((3x + 1)(x + 2)\).
This practiced method gives a clear pathway to factorization, simplifying even the most intimidating polynomials.

Canceling Common Factors

Canceling common factors is a crucial step in simplifying rational expressions. Here, the goal is to find terms that can be eliminated from both the numerator and the denominator, which simplifies the expression to its lowest terms. This technique is not only necessary for simplifying complex fractions but also helps in gaining correct solutions efficiently.
In our example, after factoring, the expression is \((3x + 1)(x + 2)\) for the numerator and \((3x + 1)(x + 4)\) for the denominator. Notice that the term \((3x + 1)\) appears in both the numerator and the denominator. Hence, it can be canceled out.
Once these common factors are eliminated, the remaining expression is \(\frac{x + 2}{x + 4}\). This is much simpler and shows the power of canceling: a bulky fraction turns into a far more concise expression by removing identical terms from both parts of the fraction.
Always ensure the factors you cancel are the same in both parts, as leaving different but similar-looking terms can lead to incorrect solutions.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contains an algebraic expression. These expressions can look quite complex at first, but simplifying them often makes them easier to work with.
Working with algebraic fractions follows the same basic principles as simplifying numeric fractions, such as finding common factors and reducing the fraction to its simplest form.
Consider the fraction \(\frac{3x^2 + 7x + 2}{3x^2 + 13x + 4}\). Before simplifying, both the numerator and denominator are expressed as polynomials, which are then factored to reveal common factors.
After factoring, you're able to cancel the common terms, just as you would with numeric fractions. The result is the simplified expression \(\frac{x + 2}{x + 4}\).
These steps can be applied to different algebraic fractions, making what initially seems complicated much more manageable. Keeping these principles in mind makes algebraic operations and fraction simplifications straightforward and effective.