Question

Multiply \(\frac{x^2-2 x-3}{x^2+3 x} \cdot \frac{x^2-9}{x^2+2 x+1}\). Write your answer in lowest terms. A. \(\frac{2(x-1)(x-3)}{x(x+1)^2}\) B. \(\frac{(x-3)^2}{x(x+1)}\) C. \(\frac{2(x-3)}{x(x+1)}\) D. \(\frac{(x-1)(x-3)^2}{x(x+1)^2}\)

Short answer

Option D

Step-by-step solution

- Factorize each polynomial

Factorize the numerator and the denominator of both fractions. For \(\frac{x^2-2x-3}{x^2+3x}\), the numerator can be factorized as \(x^2-2x-3 = (x-3)(x+1)\) and the denominator as \(x^2+3x = x(x+3)\). For \(\frac{x^2-9}{x^2+2x+1}\), the numerator can be factorized as \(x^2-9 = (x-3)(x+3)\) and the denominator as \(x^2+2x+1 = (x+1)^2\).

- Rewrite the expression with factored polynomials

Rewrite the original expression using the factored forms: \[\left(\frac{(x-3)(x+1)}{x(x+3)}\right) \cdot \left(\frac{(x-3)(x+3)}{(x+1)^2}\right)\]\.

- Simplify by canceling common factors

Cancel the common factors in the numerators and denominators. The common factors are \(x+3\), \(x+1\), and \(x-3\): \[\frac{(x-3)(x+1)}{x(x+3)} \cdot \frac{(x-3)(x+3)}{(x+1)^2} = \frac{(x-3)(x-3)}{x(x+1)} = \frac{(x-3)^2}{x(x+1)^2}\].

- Compare the simplified result with the given options

Compare the simplified form \(\frac{(x-3)^2}{x(x+1)^2}\) with the given choices to find that it matches option D.

Key concepts

Factoring Polynomials

Factoring polynomials is a key skill in algebra. It involves writing a polynomial as a product of its simpler factors. This can make complex expressions easier to manage and solve. To factor polynomials, start by identifying common factors and using algebraic identities.

For example, consider the quadratic polynomial:\[x^2 - 2x - 3\]You can factorize it by finding two binomials that multiply to give the original quadratic. Here, it can be written as:\[(x-3)(x+1)\]This means the values \(-3 \) and \(1 \) are roots of the polynomial.

In our problem, we have multiple polynomials to factor. For \((x^2-2x-3)\) we get \((x-3)(x+1)\), and for \((x^2+3x)\), we get \((x)(x+3)\). Similarly, for \((x^2-9)\), we use the difference of squares formula, resulting in \((x-3)(x+3)\), and for \((x^2+2x+1)\), we use perfect square trinomial, giving \((x+1)^2\).

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This involves canceling out common factors from the numerator and the denominator.

After factoring, our expression looks like this:\[\frac{(x-3)(x+1)}{x(x+3)} \times \frac{(x-3)(x+3)}{(x+1)^2}\]Now, we can simplify by canceling common terms present both in the numerator and the denominator. Cancel the common factors \(x+3\) and \(x+1\), which leaves us with:\[\frac{(x-3)(x-3)}{x(x+1)}\]This further simplifies to:\[\frac{(x-3)^2}{x(x+1)^2}\]through simplification. Notice that the remaining fraction is now in its simplest form because there are no common factors left to cancel.

Algebra

Algebra involves the manipulation of symbols and numbers to solve equations and understand relationships between quantities. Mastering the basic operations, such as addition, multiplication, and division, is crucial.

In this exercise, we have applied algebraic rules to manipulate and simplify rational expressions. Here’s the step-by-step breakdown:

• We started by factoring the polynomials to rewrite complex fractions into simpler multiplicative forms.
• Then, we rewrote the expression using factored forms to set up for simplification.
• The key to simplifying the expression was recognizing and canceling common factors correctly.

Multiplying these rational expressions requires careful navigation to avoid common mistakes, like failing to factor correctly or missing out on cancelable terms.

Ultimately, through algebraic manipulation and simplification, we derived the solution:\(\frac{(x-3)^2}{x(x+1)^2}\), which matched option D in the original problem’s multiple-choice answers.