Problem 26 - Algebra and Trigonometry

Question

\(21-34\) . Perform the multiplication or division and simplify. $$ \frac{x^{2}-x-6}{x^{2}+2 x} \cdot \frac{x^{3}+x^{2}}{x^{2}-2 x-3} $$

Step-by-Step Solution

Answer

The simplified expression is \(x\).

Step 1: Factorize the Numerator of the First Fraction

The numerator of the first fraction is \(x^2 - x - 6\). We need to factorize it. We look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\), so the factorized form is \((x - 3)(x + 2)\).

Step 2: Factorize the Denominator of the First Fraction

The denominator of the first fraction is \(x^2 + 2x\). We can factor out an \(x\), giving us \(x(x + 2)\).

Step 3: Factorize the Numerator of the Second Fraction

The numerator of the second fraction is \(x^3 + x^2\). We can factor out an \(x^2\), resulting in \(x^2(x + 1)\).

Step 4: Factorize the Denominator of the Second Fraction

The denominator of the second fraction is \(x^2 - 2x - 3\). We look for two numbers that multiply to \(-3\) and add to \(-2\). These numbers are \(-3\) and \(1\), so the factorized form is \((x - 3)(x + 1)\).

Step 5: Set Up the Problem with Factorizations

Now substitute the factorized terms back into the original expression:\[\frac{(x - 3)(x + 2)}{x(x + 2)} \cdot \frac{x^2(x + 1)}{(x - 3)(x + 1)}\]

Step 6: Cancel Common Factors

Now, we cancel out the common factors across the numerators and the denominators. The common factors \((x - 3)\), \((x + 2)\), and \((x + 1)\) can be canceled:\[\frac{x^2}{x}\]

Step 7: Simplify the Expression

Simplify the expression \(\frac{x^2}{x}\). When \(x\) is not zero, this simplifies to \(x\). Thus, the simplified form of the initial multiplication is \(x\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra. It is the process of writing a polynomial as a product of its factors. Factors are simpler polynomials that, when multiplied together, give the original polynomial. In many algebraic problems, factorization is a necessary step to simplify expressions or solve equations easily.

For example, to factor the polynomial \( x^2 - x - 6 \), we look for two numbers that multiply to \(-6\) (the constant term) and add to \(-1\) (the coefficient of the linear term). These numbers are \(-3\) and \(2\). Therefore, the factorization is \( (x - 3)(x + 2) \).

The factoring of the polynomial \( x^2 + 2x \) involves identifying the greatest common factor (GCF). Here, the GCF is \(x\), so the factored form is \( x(x + 2) \). Factoring techniques often used include finding the GCF, applying special formulas like the difference of squares, and using the quadratic trinomial method.

Rational Expressions

A rational expression is an expression that can be written as a fraction, where the numerator and the denominator are both polynomials. Simplifying rational expressions involves canceling out common factors.

In the problem, we have the rational expression \( \frac{x^2-x-6}{x^2+2x} \cdot \frac{x^3+x^2}{x^2-2x-3} \). To simplify this, we first factor each part. Then, we look across the fractions to cancel common factors in the numerators and denominators.

The expression reduces to \( \frac{x^2}{x} \) by canceling out common terms like \((x - 3)\), \((x + 2)\), and \((x + 1)\). When simplifying rational expressions, always remember to factor first, then cancel, taking care not to cancel terms mistakenly that are not common across the numerator and the denominator.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and arithmetic operations. They allow us to formulate equations, model real-world situations, and perform calculations.

The simplified result of this exercise, \( x \), started as a complex algebraic expression but was simplified using polynomial factorization and rational expression simplification techniques.

Working with algebraic expressions often involves several steps: identifying patterns or known forms, factoring when necessary, and simplifying terms through operations such as addition, subtraction, or multiplication. Keeping track of every simplification step is essential to achieve the correct final result, in this case, reducing the initial complicated expression down to \( x \).

Mastering the simplification of algebraic expressions develops critical thinking and problem-solving skills, as it requires understanding the structure and relationships within these mathematical statements.