Question

Lainey is simplifying the following rational expression. she decides to factor the numerator first. What should she do?

(x^2 -5x+50)/x-10

A. Divide everything by 5
B. Keep change flip
C. Look for two numbers that multiply to 50 and add to -5
D. Use the differences of two squares identity

Answer 1

To factor the numerator \( x^2 - 5x + 50 \) of the rational expression \(\frac{x^2 - 5x + 50}{x - 10}\), Lainey should examine the quadratic \( x^2 - 5x + 50 \).

The goal of factoring is to express it in the form \((x - a)(x - b)\), where \(a\) and \(b\) are numbers that multiply to the constant term (50) and add to the linear coefficient (-5).

Now, let's analyze the choices given:

A. **Divide everything by 5** - This doesn't help with factoring.

B. **Keep change flip** - This pertains to dividing fractions and is not relevant in factoring.

C. **Look for two numbers that multiply to 50 and add to -5** - This is correct. We need to find two numbers that meet the requirements for factoring.

D. **Use the differences of two squares identity** - This applies to expressions like \(a^2 - b^2\) and does not apply to this quadratic expression.

Therefore, the best step Lainey should take is:

**C. Look for two numbers that multiply to 50 and add to -5.**

However, it's worth noting that no two integers satisfy this condition (as the quadratic has a positive discriminant and may indicate complex roots). Thus, Lainey might find that the expression cannot be factored over the integers.