Problem 41 - Algebra and Trigonometry

Question

Perform the multiplication or division and simplify. $$ \frac{\frac{x^{3}}{x+1}}{\frac{x}{x^{2}+2 x+1}} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^3 + x^2 \).

Step 1: Simplify the Denominators

The denominator \( x^2 + 2x + 1 \) can be factored as \( (x+1)^2 \). This simplifies the fraction \( \frac{x}{x^2 + 2x + 1} \) to \( \frac{x}{(x+1)^2} \).

Step 2: Rewrite the Division as Multiplication

Rewriting the expression \( \frac{\frac{x^3}{x+1}}{\frac{x}{(x+1)^2}} \) as multiplication, we get \( \frac{x^3}{x+1} \cdot \frac{(x+1)^2}{x} \).

Step 3: Cancel and Simplify

Cancel out terms that appear both in the numerator and the denominator. Here, \( x \) from \( \frac{x^3}{x+1} \) can cancel with the \( x \) in the denominator. Similarly, one \( x+1 \) can cancel out from \( \frac{(x+1)^2}{x} \). This simplifies to \( x^2 \cdot (x+1) = x^2 (x+1) \).

Step 4: Final Result

Multiply the remaining terms: \( x^2 \cdot (x+1) \), which results in \( x^3 + x^2 \).

Key Concepts

Factoring

Factoring involves breaking down an expression into simpler parts, known as factors, which can be multiplied together to obtain the original expression. It's a core concept in algebra, particularly useful in simplifying expressions and solving equations. In the context of polynomials, factoring helps in expressing them as a product of simpler polynomials. For example, consider the polynomial expression in the denominator of our original problem: - The expression, \( x^2 + 2x + 1 \) can be factored to \((x+1)^2\).This means that \( x^2 + 2x + 1 \) is equivalent to multiplying \((x+1)\) by itself. Recognizing this can drastically simplify your work, particularly when you're dealing with division or multiplication of fractions because certain factors can be canceled out with factors in the numerator.Generally, here are some common forms to recognize that can help with factoring:

• Difference of squares: \( a^2 - b^2 = (a-b)(a+b) \)
• Perfect square trinomials: \( a^2 \, \pm \, 2ab \, + \, b^2 = (a \, \pm \, b)^2 \)
• Trinomials: \( ax^2 + bx + c \)

Understanding these forms can save you a lot of effort when working through algebraic operations.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions where the numerators and denominators are polynomials. This process uses factoring, as it enables us to recognize and cancel out common terms across the numerator and the denominator. Our goal is to rewrite the expression in its simplest form. Looking at the example problem:- Originally, we have the expression \( \frac{\frac{x^3}{x+1}}{\frac{x}{(x+1)^2}} \).To simplify this, you need to multiply by the reciprocal of the denominator. This converts the expression to a multiplication problem:- Changed to \( \frac{x^3}{x+1} \times \frac{(x+1)^2}{x} \).Once it is rewritten, the simplification process involves canceling out common factors:- The \(x\) in the numerator cancels out with \(x\) in the denominator.- One \((x+1)\) in the denominator cancels with one in the numerator.The simplified form emerges as \( x^2 \times (x+1) \). By eliminating terms in both the numerator and the denominator, the expression becomes cleaner and easier to work with in further calculations.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.A polynomial can have several terms, each of which is composed of a constant multiplied by one or more variables raised to integer powers. The highest power of the variable in the polynomial is called the degree of the polynomial.For example, in our exercise, the expression can be viewed as the multiplication of two simplified polynomials, \( x^2 \) and \( (x+1) \).Understanding polynomials involves knowing how to:

• Add, subtract, and multiply polynomials.
• Factor polynomials into simpler "building blocks".
• Identify the degree and leading coefficient of a polynomial.

In our simplified result of \( x^3 + x^2 \), each term represents a product of coefficients (which are number parts) and variable powers (which raise the variable to some exponent). Mastering polynomials is fundamental in algebra as they form the building blocks of more complex algebraic expressions and equations. By fostering a strong understanding of polynomials, students can unlock a wide range of mathematical applications.