Reducing an operation

[microvelo]

Hello, I'm wondering how I would go about reducing this operation?

- ( 6 / x[sup:2qkasp20]2[/sup:2qkasp20] - 8x + 12) + ( x / x[sup:2qkasp20]2[/sup:2qkasp20] - 36) - ( x / x[sup:2qkasp20]2[/sup:2qkasp20] + 4x -12)

I could factor all of the values in the denominator but then how do I go from there?

Thanks in advanced!

 

[mmm4444bot]

Hi microvelo,

We type the grouping symbols around the denominators, instead.

-6/(x^2 - 8x + 12) + x/(x^2 - 36) - x/(x^2 + 4x - 12)

Yes, factor each of the denominators.

You need to see the factors, in order to determine the Least Common Denominator (LCD). Then write equivalent ratios so that each has the LCD as its denominator.

Once you have a common denominator on all three ratios, you then combine them by adding the numerators together and writing that sum over the LCD.

Do you remember how to add Rational numbers ? The steps are exactly the same; the only difference is that your exercise contains symbolic numbers, like x^2 - 36 and -x, instead of only constants.

Business is slow this morning, so I'll type up two examples for you. The first one shows how to add Rational numbers (in case you forgot).

EG:

1/15 + 1/20

Before we can add these two Rational numbers, each of them must be re-expressed with a denominator that is common to both.

We factor what we can in the denominators.

1/(3*5) + 1/(4*5)

The LCD is 3*4*5, which is 60

We multiply 1/(3*5) by 4/4 because it needs a factor of 4 in the denominator, to get the LCD.

(1*4)/(3*4*5) = 4/60

We multiply 1/(4*5) by 3/3 because it needs a factor of 3 in the denominator, to get the LCD.

(1*3)/(3*4*5) = 3/60

Now that each ratio has the same denominator, we can combine them into a single ratio by adding the numerators.

1/15 + 1/20 = 4/60 + 3/60 = (4 + 3)/60 = 7/60


Next, I'll show an example that combines two algebraic ratios with quadratic denominators (as in your exercise). Try to "see" the same pattern of steps as above.

EG:

-6/(x^2 - 1) - x/(x^2 + 3x + 2)

We need a common denominator, before we can combine these two ratios into one. We want the LCD.

We factor what we can in the denominators.

-6/[(x + 1)(x -1)] - x/[(x + 1)(x + 2)]

The LCD is (x + 1)(x - 1)(x + 2)

We write equivalent ratios, to make each denominator the LCD.

The first ratio -6 over (x + 1)(x - 1) needs a factor of (x + 2) in its denominator. Therefore, we multiply the ratio by (x + 2)/(x + 2).

That gives us (-6)(x + 2) over (x + 1)(x - 1)(x + 2)

The second ratio -x over (x + 1)(x + 2) needs a factor of (x - 1) in its denominator. Therefore, we multiply it by (x - 1)/(x - 1).

That gives us (-x)(x - 1) over (x + 1)(x + 2)(x - 1)

Now both denominators are the same, so we can combine numerators, writing their sum over the LCD.

(-6)(x + 2) - x(x - 1) over (x + 1)(x - 1)(x + 2)

Simplifying this numerator gives:

(-x^2 - 5x - 12)/[(x + 1)(x - 1)(x + 2)]

Some people say that it's better to factor -1 out of the numerator and move it in front of the ratio.

-(x^2 + 5x + 12)/[(x + 1)(x - 1)(x + 2)]

Some people say that you should expand the LCD (i.e., multiply its factors together) and write the denominator as a cubic polynomial. I say NO. Leave it in factored form because when you start using these simplified ratios in later math topics, you'll want to be able to see the factors on the bottom, for potential cancellations involving the numerators of other ratios.



Time for me to stop talking.

Is this enough information for you to follow the same pattern and combine equivalent ratios together, in your exercise ?

Please show any work that you can, if you need more help.

Cheers ~ Mark