Write each rational expression in lowest terms. 8x² + 16 / 4x²

Verified step by step guidance

1. Identify the rational expression given: \(\frac{8x^2 + 16}{4x^2}\).
2. Factor the numerator where possible. Notice that both terms in the numerator share a common factor of 8: \$8x^2 + 16 = 8(x^2 + 2)$.
3. Rewrite the expression using the factored numerator: \(\frac{8(x^2 + 2)}{4x^2}\).
4. Simplify the fraction by dividing the common factors in the numerator and denominator. Since 8 and 4 share a common factor, divide both by 4: \(\frac{8}{4} = 2\), so the expression becomes \(\frac{2(x^2 + 2)}{x^2}\).
5. Express the simplified rational expression clearly as \(\frac{2(x^2 + 2)}{x^2}\), which is the expression in lowest terms.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and denominator are polynomials. Simplifying rational expressions involves factoring and reducing common factors, similar to simplifying numerical fractions.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into a product of simpler polynomials or constants. Recognizing common factors, such as greatest common factors (GCF), is essential to simplify expressions effectively.

Simplifying Fractions

Simplifying fractions means dividing the numerator and denominator by their greatest common factor to reduce the expression to its lowest terms. This process ensures the expression is easier to work with and understand.