Problem 44 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{x^{2}+6 x+9}{2 x^{2}+6 x} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x+3}{2x}\), with restrictions \(x \neq 0, -3\).

Step 1: Factor the Numerator

The numerator of the expression is a trinomial: \(x^2 + 6x + 9\). This can be factored by recognizing it as a perfect square trinomial. The pattern \((a+b)^2 = a^2 + 2ab + b^2\) can be applied, where \(a = x\) and \(b = 3\). Thus, the numerator factors to \((x + 3)^2\).

Step 2: Factor the Denominator

The denominator is \(2x^2 + 6x\). Start by factoring out the greatest common factor, which is 2x, resulting in \(2x(x + 3)\).

Step 3: Simplify the Expression

Now that both the numerator and the denominator are factored, the expression is \(\frac{(x+3)^2}{2x(x+3)}\). Since \(x+3\) is a common factor in the numerator and the denominator, it can be canceled out, assuming \(x eq -3\) to avoid division by zero. Thus, the simplified expression is \(\frac{x+3}{2x}\).

Step 4: State Restrictions

When simplifying rational expressions, it's important to state any restrictions on the variable that would make the denominator zero. Originally, the denominator was \(2x^2 + 6x = 2x(x+3)\), so \(x eq 0\) and \(x eq -3\) to avoid division by zero.

Key Concepts

Factoring Trinomials

Factoring trinomials is a crucial step when simplifying rational expressions, especially when dealing with quadratic polynomials. A trinomial is a polynomial with three terms, commonly in the form \( ax^2 + bx + c \). We often encounter it in problems requiring simplification.

In the exercise, the numerator \( x^2 + 6x + 9 \) is a trinomial. To factor it, we recognize it as a perfect square trinomial. This means it takes the form of \((a+b)^2 = a^2 + 2ab + b^2\).

• a: the term that squares to give \( a^2 \) is \( x \).
• b: the constant term when squared gives \( b^2 \). Here, \( b = 3 \) since \( 3^2 = 9 \).

So, we factor it as \((x+3)^2\).

Recognizing patterns, like perfect squares, cues you in on quicker factorization techniques. Practice factoring by identifying such forms along with differences of squares or sum-products.

Simplifying Fractions

Simplifying fractions involves reducing a fraction into its simplest form. For rational expressions, this means canceling common factors from the numerator and denominator.

For example, in the expression \(\frac{(x+3)^2}{2x(x+3)}\), we identify \((x+3)\) as a common term that appears in both the numerator and the denominator.

• To simplify, you cancel out \(x+3\) if it's not equal to zero. Hence you are left with \(\frac{x+3}{2x}\).

One must be cautious to avoid canceling terms that would lead to undefined expressions - division by zero is one such concern. Be sure to revisit the domain restrictions after simplifying.

Remember, to maintain mathematical accuracy, state restrictions such as \(x eq 0\) and \(x eq -3\), derived from terms in the denominator before simplification.

Polynomial Division

Polynomial division is a process to simplify expressions involving polynomials by dividing one polynomial by another.

In our example, the polynomial expression \(\frac{x^2+6x+9}{2x^2+6x}\) simplifies through factoring each part before division. The numerator and the denominator both provide factors that can be divided out, simplifying the expression.

• Numerator: The trinomial \(x^2 + 6x + 9\) becomes \((x+3)^2\).
• Denominator: The term \(2x^2 + 6x\) factors to \(2x(x+3)\).

Through division, simplify it to the rational form \(\frac{x+3}{2x}\).

Being able to skillfully divide polynomials might appear daunting at first, but mastering factoring trivializes the process. Always revisit your factorization step if simplifications don't seem to align. This process will assure accuracy while also uncovering any hidden disallowed values in your simplification.