Problem 69 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{2 a^{2}+4 a+1}{a^{2}+2 a+3} $$

Step-by-Step Solution

Answer

Answer: The simplified expression for the given function is $$\frac{2a^2 + 4a + 1}{a^2 + 2a + 3}$$. There are no domain restrictions for the variable 'a', since the denominator has no real roots.

Step 1: Factor the numerator and denominator

In this step, we will try to factor the numerator and the denominator to find if there is any common factor that can be canceled out. However, both the numerator and the denominator are not factorable. So, we can't simplify the expression by factoring. We can move to the next step.

Step 2: Check for any domain restrictions

We will check for any values of 'a' that would make the denominator equal to zero, as these values would make the fraction undefined. We have: $$ a^2 + 2a + 3 = 0 $$ This is a quadratic equation, and we can find its roots using the discriminant. The discriminant is given by: $$ \Delta = b^2 - 4ac $$ Where, `a`, `b` and `c` are the coefficients of the quadratic equation In this case, we have a = 1, b = 2, and c = 3. So, $$ \Delta = 2^2 - 4(1)(3) = 4 - 12 = -8 $$ Since the discriminant is negative, the quadratic equation has no real roots. Hence, there are no restrictions on the value of 'a'.

Step 3: Write the final answer

As we couldn't simplify the expression and found no domain restrictions for the value of 'a', the final answer is simply the given expression: $$ \frac{2a^2 + 4a + 1}{a^2 + 2a + 3} $$

Key Concepts

Rational Expressions

Rational expressions are like fractions but with polynomials in the numerator and denominator. They come in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. It's essential to understand that just like regular fractions, we can only simplify rational expressions by canceling factors, meaning terms that multiply together to create the polynomials.

However, before you go on simplifying those expressions, always check if the numerator and denominator can be factored. If common factors exist, they can be canceled, simplifying the expression. Sometimes, as in the current exercise, neither the numerator nor the denominator can be factored which means the expression remains as is.

Rational expressions can be diverse and include subtracting, adding, multiplying, or dividing them. Each operation has specific steps that one needs to follow carefully while considering the expression's whole structure.

Quadratic Equations

A quadratic equation is any equation that can be written in the standard form \( ax^2 + bx + c = 0 \). These equations are prevalent in algebra due to their unique properties and the wealth of methods available for solving them.

One such method involves using the discriminant, \( \Delta = b^2 - 4ac \). The discriminant lets us understand how many and what type of solutions a particular quadratic equation has:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), there's exactly one real root or a repeated real root.
• If \( \Delta < 0 \), there are no real roots, only complex ones.

In our case study, the discriminant calculation yields a negative result, meaning there are no real roots. This helps us establish domain restrictions for rational expressions, as discussed earlier.

Knowing these factors aids in understanding the nature of quadratic equations and how these characteristics affect other algebraic processes.

Domain Restrictions

When dealing with rational expressions, domain restrictions are critical. While fractions can result in undefined expressions when their denominators equal zero, rational expressions can face similar issues.

In algebra, the domain of a rational expression is all the possible input values (often represented by \( x \) or any other variable) except for those which make the denominator zero. To find these "restricted" values, solve the equation formed by setting the denominator equal to zero. These solutions are the domain restrictions.

However, using the quadratic equation \( a^2 + 2a + 3 = 0 \) from the presented exercise, we find that no real numbers make the denominator zero, evidenced by the negative discriminant. Thus, there are no domain restrictions for this particular expression.

Having a systematic approach to locating these domain restrictions ensures that while working with rational expressions, the resulting values remain valid and defined within their algebraic context.