Rational Expression 3 | Math Is Still Fun

Changing the Denominator!

Fasten your safety belts! It's less fun that way!

This is an example of one of the reasons I hated math in high school. They throw in fancy terminology to memorize, along with a complicated way of saying something boring. The icing on top is the fact that I cannot fathom a use in my head where this is necessary. That was teenage me and still is me.

These guys are clowns, pull off their scary mask. This is how to solve the problem. After that we have to see why this was necessary to learn.

Let us keep things simple, now, and think about what we are being asked and how we are going to answer.

First, they want us to

/clearthroat

Transform the bestowed utterance into an expression of equal grandeur.....

TRANSLATED TO HILL BILLY

Y'all reckon y'all could kindly transform this here numbers and letters they call them variables into a good ol' hillbillY readin' that is the same text but different letters and numbers, I guess? What am I asking? Much obliged!

TRANSLATED TO SMART PEOPLE

Make the given expression an equivalent expression

HIT THE PAUSE BUTTON RIGHT THERE

What did that dude just say again? Too many words that begin with 'e' and then Expresso is not on the menu.

From what I can gather in this passage,

In an exerted effort to sound intelligent, one man took....

Take this and make it look different.

...and then bestowed upon humanity in elegant scripture....

Make the thing I give you, into something new but the same.

...and then another genius shed forth....

Make the given expression an equivalent expression.

Then he was the last guy, and everyone still hates him.

I have a wheel, it is blue. I paint the blue wheel with the color red. The wheel is still a wheel, but now it is red.

Just start simple.

You have been given a fraction. A/B

You have been given another equation, we will call it C.

The question wants you to take the fraction's denominator, B, and make it of equal value to C.

To make B = C is the answer, and we need to pre-req our journey here.

1. If we do anything to B, we have to do the same thing to A in order to keep the value correct.

Okay, now let us make B and C equal each other.

How do we make x + 8 be equal in value to x³ + 8x²?

We just introduced the exact equations for B and C. Stop trying to think about anything but the letter B and C. One is an equation, and the other is an equation. One is the letter B. The other is the letter C.

To get the letter B to be equal to the letter C, given that they are indeed already different, we need to find something D.

So that when we have B*D, we will be equal to C.

What do we multiply one by to get the other?

Let us look at C:

x³ + 8x²

First, I do not see anyway I can make this into an ( x + M ) * ( x + N ) sort of equation.

That is something they like to throw at you, so I generally begin there.

If we remove x² from x³ + 8x² we know we have then

x²( x + 8 )

That is another way of looking at the value of C.

Recall the value now of B

x + 8

By simplifying our denominator, now we easily see what we must do to answer the question!

To make B = C, we must multiply B times a D.

Such that

B * D = C

OR

C = D * B

To satisfy:

x³ + 8x² == x² * ( x + 8 )

C == D * B

with D being equal to x²

Back to the top. We want to take the fraction A/B and change it into a new fraction P / C. Now that we know how to modify B to become equal value with C, we must perform the same function to the numerator. The same function hat was performed to change B into C. We must now do this to A.

Recalling from the example problem, we see that A is equal to

x - 7

To make B equal C, we had to multiply B times D. So now we must multiply A times D to keep the fraction correct value.

( A ) * ( D ) = P

( x + 7 ) * ( x² )

Using our distribution property rules:

(x² *x) + (x² * 7)

Which yields:

x³ + 7x²

Now, taking numerator and placing over denominator: