Question

Challenge Problems $$\frac{x^{3}-a^{2} x}{x^{2}-2 a x+a^{2}}$$

Short answer

\[\frac{x(x+a)}{x-a}\]

Step-by-step solution

Factor the Numerator

Factor out an x from the numerator to simplify the expression. This gives us: \[\frac{x(x^{2}-a^{2})}{x^{2}-2ax+a^{2}}\]

Factor the Denominator

Observe that the denominator is a perfect square trinomial. Factor the denominator to get: \[\frac{x(x^{2}-a^{2})}{(x-a)^{2}}\]

Factor the Difference of Squares in the Numerator

Factor the difference of squares in the numerator: \[x^2 - a^2 = (x+a)(x-a)\]. So the factored form of the numerator is: \[x(x+a)(x-a)\]

Simplify the Fraction

Cancel out the common terms (x-a) in the numerator and the denominator: \[\frac{x(x+a)(x-a)}{(x-a)(x-a)} = \frac{x(x+a)}{x-a}\]

Final Simplification

Since there no longer are any common factors, the simplified form of the expression is: \[\frac{x(x+a)}{x-a}\]

Key concepts

Difference of Squares

The difference of squares is a mathematical concept that applies to algebra, particularly in factoring polynomials. It describes the subtraction between two squared numbers, which can be factored into a product of two linear binomials. The general form of a difference of squares is \( a^2 - b^2 \) and it can be factored as \( (a + b)(a - b) \).

For instance, if we have \( x^2 - a^2 \), it signifies the difference between the square of \( x \) and the square of \( a \). By using the difference of squares formula, we can factor \( x^2 - a^2 \) into \( (x + a)(x - a) \). This method is especially helpful when dealing with complex algebraic expressions such as the ones found in the challenge problem provided.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be written as the square of a binomial. In general, a perfect square trinomial has the form \( a^2 + 2ab + b^2 \), which is equivalent to \( (a+b)^2 \). It's made up of three terms: the squares of two terms and twice the product of the two terms.

One of the keys to recognizing a perfect square trinomial is to note the relationship between its terms. For example, the expression \( x^2 - 2ax + a^2 \) represents a perfect square trinomial since \( a^2 \) is the square of \( a \) and \( 2ax \) is twice the product of the \( x \) and \( a \). When we factor a perfect square trinomial, the result is a binomial squared. In our exercise, \( x^2 - 2ax + a^2 \) factors to \( (x-a)^2 \).

Simplifying Rational Expressions

Simplifying rational expressions involves rewriting them in the most reduced form by eliminating common factors from the numerator and the denominator. The key steps are to factor both the numerator and the denominator and then to reduce any common factors that appear in both.

For example, when simplifying the expression \( \frac{x^3 - a^2 x}{x^2 - 2ax + a^2} \), we first factor out an \( x \) from the numerator and recognize the denominator as a perfect square trinomial \( (x-a)^2 \). After factoring, we look for common terms in the numerator and the denominator that can be canceled out. By canceling out the common \( x-a \) terms, we are simplifying the rational expression to its lowest terms, resulting in the final simplified expression \( \frac{x(x+a)}{x-a} \).