Question

Divide and reduce. Try some by calculator. $$\frac{1}{x^{2}+17 x+30} \div \frac{1}{x+15}$$

Short answer

\(\frac{1}{x+2}\)

Step-by-step solution

Understand Division of Fractions

To divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction flips the numerator and the denominator. In this case, the reciprocal of \(\frac{1}{x+15}\) is \(x+15\).

Multiply by the Reciprocal

Multiply \(\frac{1}{x^2+17x+30}\) by the reciprocal of \(\frac{1}{x+15}\) which is \(x+15\).

Perform the Multiplication

After multiplying, the equation is \(\frac{1}{x^2+17x+30} \times (x+15)\). When multiplying, you can multiply the numerators together and the denominators together. Since our numerator is 1, the multiplication simplifies to \(\frac{x+15}{x^2+17x+30}\).

Factor the Denominator

Factor the quadratic expression in the denominator. Factoring \(x^2+17x+30\) gives us \(x+2\) and \(x+15\) as the factors. So, the denominator factors to \(\frac{x+15}{(x+2)(x+15)}\).

Simplify the Expression

Cancel the common factors in the numerator and denominator. The factor \(x+15\) is common to both and gets cancelled out, leaving us with \(\frac{1}{x+2}\).

Key concepts

Reciprocal of a Fraction

Understanding the concept of the reciprocal of a fraction is essential for dividing fractions in algebra. The reciprocal is quite simply the fraction flipped upside down. That is to say, the numerator becomes the denominator and the denominator becomes the numerator. For example, the reciprocal of \( \frac{1}{x+15} \) is \( x+15 \), which is essentially \( \frac{x+15}{1} \).

When dividing fractions, instead of performing traditional division as you might with whole numbers, you multiply by this reciprocal. Knowing how to find the reciprocal quickly is a useful skill because it simplifies the division of fractions to multiplication, a more straightforward operation.

Multiplying Fractions

Once you've determined the reciprocal of a fraction, multiplying fractions becomes the next step in the division process. When multiplying fractions, it's a matter of performing two simple multiplications: numerator times numerator and denominator times denominator.

The trick to making this process more manageable is keeping an eye out for any factors that might cancel out, simplifying the multiplication. However, it's essential not to cancel any factors that are being added or subtracted—this mistake is a common pitfall. In our example, we end up multiplying \(1 \times (x+15) \) and \( (x^{2}+17x+30) \times 1 \) which simplifies beautifully, since any number times 1 is itself.

Factoring Quadratic Expressions

Factoring quadratic expressions is a fundamental skill in algebra, particularly when simplifying algebraic expressions. A quadratic expression is typically in the form \( ax^2 + bx + c \) and factoring involves breaking it down into a product of two binomials.

In our textbook exercise, the quadratic expression \(x^2 + 17x + 30\) is factored to find its roots. The process involves finding two numbers that both add to give 17 (the coefficient of \(x\)) and multiply to give 30 (the constant term). After a bit of trial and error, we determine that those two numbers are 2 and 15, leading us to the factors \(x+2\) and \(x+15\).

Factoring can simplify the division of fractions significantly because it allows us to cancel out common factors in the numerator and denominator, thus simplifying the overall expression.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is often the final step in solving algebra problems, and being proficient at it can make a world of difference. After factoring and performing operations like multiplication or division, you should always look to simplify the expression.

Simplification may involve canceling out common factors in the numerator and denominator, as we saw in our exercise. When we reached \( \frac{x+15}{(x+2)(x+15)} \), we noticed that \(x+15\) was a common factor and could be canceled out, ultimately simplifying the expression to \( \frac{1}{x+2} \).

Remember that you can only cancel factors that are being multiplied together, not added or subtracted. Also, check that the expression is completely factored before canceling anything; missing a factor can lead to an incorrect solution.