Problem 85 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{3 x^{2}-21 x+18}{x^{2}+5 x+6} \div(x+2) $$

Step-by-Step Solution

Answer

Answer: The result of the division of the given algebraic expression is $$\frac{3(x-2)(x-3)}{(x+2)^{2}(x+3)}$$.

Step 1: Factorizing the expression

We need to factorize the numerator, denominator of the fraction, and the monomial divisor if possible: $$ \frac{3 x^{2}-21 x+18}{x^{2}+5 x+6} \div(x+2) = \frac{3(x-2)(x-3)}{(x+2)(x+3)} \div(x+2) $$

Step 2: Multiplying by the reciprocal of the divisor

Now, to perform the division, we will multiply the given expression by the reciprocal of the divisor \((x+2)\): $$ \frac{3(x-2)(x-3)}{(x+2)(x+3)} \cdot \frac{1}{x+2} = \frac{3(x-2)(x-3)}{(x+2)(x+3)(x+2)} $$

Step 3: Simplifying and finding the solution

Now, we can simplify the expression by canceling out any common factors: $$ \frac{3(x-2)(x-3)}{(x+2)(x+3)(x+2)} = \frac{3(x-2)(x-3)}{(x+2)^{2}(x+3)} $$ The simplified expression is our final answer: $$ \frac{3(x-2)(x-3)}{(x+2)^{2}(x+3)} $$

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step when dealing with algebraic expressions, particularly when working with rational expressions. To factor a polynomial, you need to express it as a product of simpler polynomials. This process often involves identifying common factors, differences of squares, or employing techniques like grouping.
For instance, when faced with the polynomial \(3x^2 - 21x + 18\), you first look for the greatest common factor among the coefficients of the terms. In this case, the number 3 can be factored out, resulting in \(3(x^2 - 7x + 6)\). Further breaking down the quadratic expressions inside the parentheses involves finding two numbers that multiply to give the constant term (6) and add to give the coefficient of the linear term (-7), which are -2 and -3. Thus, \(x^2 - 7x + 6\) factors into \((x-2)(x-3)\).
Understanding how to break down complex polynomials into simpler ones allows for easier manipulation, especially in solving equations and simplifying expressions.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. They are similar to numerical fractions but involve variables. Proper manipulation and simplification of these expressions often involve factoring as well.
For the given expression \(\frac{3(x-2)(x-3)}{(x+2)(x+3)}\), the task is to simplify it or use it in operations like division by another expression. It is essential to ensure that no part of the denominator becomes zero, as a division by zero is undefined in mathematics. This type of expression also includes restrictions or constraints on the variable values. These restrictions come from the values that make the denominator zero. For \(x+2\), this value would be \(-2\), and for \(x+3\), it would be \(-3\).
Handling rational expressions correctly allows you to simplify them or combine them effectively in calculations.

Multiplication and Division of Fractions

In algebra, the multiplication and division of fractions can be applied to rational expressions to simplify or solve them.
When multiplying fractions, simply multiply the numerators together and the denominators together. With division, however, you multiply by the reciprocal of the divisor. In our example, \(\frac{3(x-2)(x-3)}{(x+2)(x+3)} \div (x+2)\) is transformed by taking the reciprocal of \(x+2\), which is \(\frac{1}{x+2}\), and then proceeding to multiply.
This results in \(\frac{3(x-2)(x-3)}{(x+2)(x+3)(x+2)}\).
Simplification is often the final step, where common factors are canceled out to produce the simplest form of the expression. Applying these operations correctly ensures that the fractions retain their mathematical integrity and provide accurate results.