Question

Find the quotient. $$ \frac{x^2-x-6}{2 x^4-6 x^3} \div \frac{x+2}{4 x^3} $$

Short answer

The quotient of the division of the given fractions is \( 4x^3 \)

Step-by-step solution

Express division as multiplication of reciprocal

Rewrite the division as multiplication by the reciprocal of the second fraction: \[ \frac{x^2 - x - 6}{2x^4 - 6x^3} \times \frac{4 x^3}{x+2} \]

Factorizes polynomials

Factorize the polynomials: \[ \frac{(x - 3) (x + 2)}{2 x^3 (x - 3)} \times \frac{4x^3}{x+2} \]

Cancel out terms

Notice that there are common terms in the numerator and the denominator of the fractions. Cancel out \( (x - 3) \) from the first fraction and \( (x + 2) \) across the fractions to simplify the equation: \[ \frac{2 x^3 \times 4x^3}{2 x^3} \]

Simplify the expression

Simplify the final expression to get the solution: \[ 4x^3 \]

Key concepts

Multiplication of Reciprocals

Understanding the multiplication of reciprocals is crucial when dividing polynomials, as it simplifies the process significantly. A reciprocal is simply a flipped version of the original fraction, meaning we exchange the numerator and the denominator. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). When dividing by a fraction, we multiply by this reciprocal.

Let's apply this idea to polynomial division. As seen in our exercise, dividing \( \frac{x^2 - x - 6}{2x^4 - 6x^3} \) by \( \frac{x+2}{4x^3} \) involves multiplying the first fraction with the reciprocal of the second. We rewrite the equation as \( \frac{x^2 - x - 6}{2x^4 - 6x^3} \times \frac{4x^3}{x+2} \), effectively turning the division into multiplication, thus simplifying the process.

Factoring Polynomials

Factoring polynomials is a powerful technique that enables us to break down complex expressions into simpler, multipliable factors. This process is useful in polynomial division as it allows us to identify and ultimately cancel common terms.

Consider the numerator \(x^2 - x - 6\); we factor it to \( (x - 3)(x + 2) \). We do this by finding two numbers that when multiplied give us the product of the coefficient of \(x^2\) and the constant term (in this case, 1 * -6), and when added, give us the coefficient of \(x\) (here, -1). These numbers are -3 and +2. Similarly, the denominator \(2x^4 - 6x^3\) factors out a \(2x^3\), leaving us with \(2x^3(x - 3)\).

Canceling Common Terms

When we have common terms in both the numerator and the denominator, we can cancel them out to simplify the expression. This is due to the principle that \( \frac{a}{a} = 1 \), provided \( a \) is not equal to zero.

In the given problem, after factoring, we observe that \( (x - 3)\) and \( (x + 2)\) appear in both the numerator and denominator. Since these are equal, we can cancel them, much like eliminating identical terms from both sides of an equation. This cancellation leads to a much simpler form and brings us closer to the final simplified result, which in this case would leave us with \(2 x^3 \times 4x^3\) in the numerator and a single \(2 x^3\) in the denominator.

Simplifying Expressions

The last and possibly the most satisfying step in polynomial division is simplifying the expression. Simplification involves combining like terms, reducing fractions, and carrying out any arithmetic operations that can simplify the polynomial even further.

After canceling the common terms in our exercise, we are left with \( \frac{2 x^3 \times 4x^3}{2 x^3} \). We can see that \(2 x^3\) in the numerator and denominator will be simplified to 1, leaving us with \(4x^3\) as the most simplified form of the expression. It's essential to always look out for these opportunities to reduce the expression to its lowest terms, ensuring the result is presented in its simplest form.