Multiply or divide, as indicated. See Example 3. ((x² + x) / 5) • 25 / (xy + y)

Verified step by step guidance

1. First, rewrite the given expression clearly as a multiplication of two fractions: \(\frac{x^2 + x}{5xy} \times \frac{25}{y}\).
2. Next, factor any expressions where possible. For example, factor \(x^2 + x\) as \(x(x + 1)\).
3. Rewrite the expression with the factored form: \(\frac{x(x + 1)}{5xy} \times \frac{25}{y}\).
4. Multiply the numerators together and the denominators together: \(\frac{x(x + 1) \times 25}{5xy \times y}\).
5. Simplify the resulting fraction by canceling common factors in numerator and denominator, such as \(x\) and any numerical factors.

Key Concepts

Algebraic Expression Simplification

Simplifying algebraic expressions involves combining like terms, factoring polynomials, and reducing fractions. This process makes complex expressions easier to work with, especially before performing multiplication or division.

Multiplication and Division of Rational Expressions

When multiplying or dividing rational expressions, factor all numerators and denominators first, then cancel common factors. Division requires multiplying by the reciprocal of the divisor to simplify the expression correctly.

Factoring Polynomials

Factoring breaks down polynomials into products of simpler polynomials or terms. Recognizing common factoring techniques, such as factoring out the greatest common factor or using special products, is essential for simplifying and manipulating expressions.