Question

Consider the rational expression x^2+6x−2/6x−5. Which statements are true about the rational expression? Select each correct answer.
The denominator has two terms.

The numerator has three terms.

6x in the numerator divides out with 6x in the denominator.

6x−56x−5 is a single factor in the denominator.

The numerator has three factors.

Answer 1

Let's first define exactly what the word term means. A term is any constant, variable, or product of constants and variables in an expression, typically separated by a + or - sign. The first two answers are about the terms in the numerator and denominator, so let's look at those.

$x^2+6x-2$ contains 3 terms - x², 6x, and -2 - so the second statement, "the numerator has three terms," is correct.

$6x-5$, the term in the denominator, has two terms - 6x and 5 - so the first statement must be correct, too.

The third statement is something you have to be careful with. It's tempting to want to cancel the 6x in the numerator and the 6x in the denominator, but the expression 6x-5 in the denominator describes a sum, not a product, so we can't cancel in the same way. For an example of why that wouldn't make sense, let's manipulate the fraction 7/8:

Since we know that 8 = 7 +1, we can rewrite the fraction as:

$\frac{7}{7+1}$

If we allowed ourselves to cancel the 7s in the numerator and the denominator, we'd suddenly end up with:

$\frac{1}{1+1}= \frac{1}{2}$

Clearly, 7/8 ≠ 1/2, and since assuming we could make that cancellation led to an absurd answer, we can safely say that - in the same way - no, we can't cancel the 6x's.

The last statement, "the numerator has three factors," rests on a result boldly called the fundamental theorem of algebra. What the fundamental theorem states is that the number of roots of any polynomial is always equal to its degree. This is the same thing as saying that the number of factors a polynomial has is equal to its highest power. The highest power of the numerator $x^2+6x-2$ is 2, which means that it has at most 2 factors, making the statement false.

So, again, here's our rational expression:

$\frac{x^2+6x-2}{6x+5}$

And here are the true statements about it:
- The denominator has two terms
- The numerator has three terms