Simplifying Rational Expressions

Front

What is a rational expression?

Back

A rational expression is a fraction where both the numerator and the denominator are polynomials.

Front

How do you simplify a rational expression?

Back

To simplify a rational expression, factor both the numerator and the denominator, then cancel any common factors.

Front

What is the standard form of a rational expression?

Back

The standard form of a rational expression is when the numerator and denominator have no common factors other than 1.

Front

What is factoring in the context of polynomials?

Back

Factoring is the process of breaking down a polynomial into simpler components (factors) that, when multiplied together, give the original polynomial.

Front

What is the difference of squares?

Back

The difference of squares is a special factoring pattern that states a^2 - b^2 = (a - b)(a + b).

Front

What is the quadratic formula?

Back

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the roots of a quadratic equation ax² + bx + c = 0.

Front

What are the steps to simplify the expression \( \frac{x^2 + 7x + 10}{x^2 - 4} \)?

Back

1. Factor the numerator: (x + 5)(x + 2). 2. Factor the denominator: (x - 2)(x + 2). 3. Cancel the common factor (x + 2). 4. Final answer: \( \frac{x + 5}{x - 2} \).

Front

What is the significance of the domain in rational expressions?

Back

The domain of a rational expression is all real numbers except where the denominator equals zero, as these values make the expression undefined.

Front

How do you identify vertical asymptotes in rational functions?

Back

Vertical asymptotes occur at the values of x that make the denominator zero, provided these values do not cancel with the numerator.

Front

What is the process to simplify \( \frac{x^2 - 4x - 45}{x^2 - 81} \)?

Back

1. Factor the numerator: (x - 9)(x + 5). 2. Factor the denominator: (x - 9)(x + 9). 3. Cancel the common factor (x - 9). 4. Final answer: \( \frac{x + 5}{x + 9} \).