Clint's Responses in WyzAnt Answers

Jackeria, 

Looks like you are working with dividing and simplifying polynomials. The problem you listed looks like (4x - 1) / (3x^2 - 11x - 26). Before trying division, we always want to check if we can simplify or factor either the numerator or the denominator. Can you? Completing the square doesn't help here, nor does the quadratic formula. Nothing nice. Looks like we're doing Polynomial long division! 

You can divide one polynomial P(x) by another Q(x) if the degree of P(x), that is, the highest power of x in P(x) is larger than the highest power in Q(x). 

In this case, P(x) = 4x - 1. Q(x) is the denominator, (3x^2 - 11x - 26) The degree of P(x) is 1, and the degree of Q(x) is 2. So we can't divide P(x) / Q(x). We could divide the reciprocal, Q(x) / P(x), then flip that to get our answer. Then we could use Polynomial Long Division.

              36x/4 - 41/16     r  -375/16

           ________________

(4x -1) | (3x^2 - 11x - 26)

           - (3x^2 - 3/4 x)

             --------------------

                      -41/4 x - 26

                    -(-41/4 x + 41/16)

                     -------------------

                                -374/16

Polynomial long division works like the long division you learned in school, except you have to keep track of the x's. Start with the largest terms, and work in. 

What factor, F, times 4x * F = 3x^2?  If we divide, we get 3x/4. That's our first term. Multiply through, then subtract. Just like regular long division. This eliminates the 3x^2 term completely, and we can move to the next term. 

Continuing in this way, we get to -375/16. We want our answer to only have positive powers of x in it, so this is our remainder. Our answer for Q(x)/P(x) is 3x/4 - 41/16  - (375/16)/(4x -1)

The reason we write (375/16) over (4x-1) is because it did not divide with 'evenly'. Just like 15/4 = 3 remainder 3/4 , or 11/2 = 5 remainder 1/2.

We are looking for P/Q, so we can just write the reciprocal (flip) of what we just found. 

So (4x - 1) / (3x^2 - 11x - 26) = 1/[ 3x/4 - 41/16 - (375/16)/(4x -1)]

Hope this helps!

Clint V. Wyzant Tutor, Norman, OK

Hi Ashley! Looks like you got a big 'ol algebra expression there. I say expression, because there's no equal signs, so it's not technically an equation.

If you are looking to simplify that expression, I can help.

You can use the distributive property here. Let's separate our expression a little:

x^2 - x    + 2x -2

Do the two terms on the left have anything in common? Yeah, they both have an x!

Let's pull that out. x^2 - x = x (x - 1)

Do the terms on the right have anything in common? Yeah, they both have a 2!

Let's pull that out. 2x - 2 = 2 (x - 1)

Now we have x^2 - x + 2x - 2 = x (x - 1) + 2 (x - 1)

Hmmm. It looks like we have some like terms here, because the terms in parentheses are the same. We can add or subtract them like any other variable.

So we have x (x -1) + 2 (x - 1) = (x + 2) (x - 1)

Does that work? We can use the FOIL method to check ourselves. First Outer Inner Last

Yep, sure does.

Hope this helps, Ashley!

Hi Laurie! When approaching this problem, ask yourself, "What is it asking?" "What objects or shapes are we dealing with?"  

We're dealing with a piece of cardboard 10" x 18". What shape is that? That's how we talk about rectangles. Length x Width. So we start with a rectangle of cardboard 18" long and 10" wide.

Now, what's with "open top box" - what does that mean? A box is a 3-dimensional rectangular solid. An open top means what? This is poorly-worded problem because we have to guess at "open top". It probably means "no lid". Can you picture or draw a box with no lid? You're going to need to do so to solve the problem.

You are being asked to find the volume of a box. What is the formula for the volume of a box?

That's right - Box Volume = Length x Width x Height = l x w x h = V

Which box? A box made out of a piece of cardboard by cutting a square out of each corner. Can you draw this?

        18 "

|-------------------------|

|                               |  10"

|-------------------------|

(Note: I drew up this problem with MS Paint, but couldn't paste the images)

Let's keep building that box.

What is the shape of the pieces we are cutting away? Check the problem - squares!

How big are they? Check the problem - size x! What does that mean? Our rectangle takes two measurements, length and a width. How many measurements do you need to draw a square? Just one.

              ___ ?____

___x___ |               |__x____     (all the squares are x tall and x wide)

|                                         |  ?

----x-----                 ----x------                       |    

            |_________|   

What does this mean? This is our picture of what our cardboard looks like after we cut the squares off it.

It looks like a cross.

Can you imagine how to make an open top box out of this? Right, just fold the sides up. Then we'd have a box with no top.

Great, now we have a box, but what is the volume? What do we need to know about the box to find the volume? Check the earlier formula!  We need to know length, width, and height!

How long is the longer base of the box? Well, the cardboard rectangle was 18" long. Then we cut away two pieces that were x long. They were "x" inches long. If you have 18 and take away two x's, how do we write that? ..... that's right, 18 - 2x = L

How long is the shorter base of the box? Well, the cardboard rectangle was 10" wide. Then we cut away two pieces that were x long. If you have 10 and take away two x's, how do we write that? ..... that's right, 10 - 2x = W

Now, how tall is the box? Hmmm. Imagine folding up those cross pieces off the ground. They stick out a distance of x inches. When something flat and x inches long is bent straight up, how far up does it go?

That's right, x inches. The box will be x inches tall, that is, a height of x inches.

Now let's put all this together. We are looking for the volume of the folded box, and volume is length x width x height. That means the volume of the box is (18 - 2x)(10 - 2x)(x). If we multiply that out (remembering our foil method), we get V(x) = 180x - 56x^2 + 4x^3.

 The final part of the problem asks, "What is the real world domain for V(x)?" What is a domain? Remember? A domain is the set of values for which the function is defined. A real world domain is the set of values for which an expression or formula makes sense.

What values make sense for x? X is a measure of what, again? The size of the squares. We cut a square out of each corner. What are the biggest and smallest squares you can cut out of our cardboard?

Here's a question: Can a box have lengths less than zero? No! That's impossible. So we need values of x for which Length (18 - 2x), Width (10 - 2x), Height (x) are greater than zero.

What are they? Can you solve them?  1) 18 - 2x > 0    2) 10 - 2x > 0    and 3) x > 0

1) x < 9 inches.   Two nine inch squares use up all the length of the cardboard -> there's not enough board to get any longer!

2) x < 5 inches for width.   Two five inch squares use up all the width.

3) x > 0 inches.  You can't have a box without some height. x needs to be greater than 0 or we just have a flat piece of cardboard.

So, are we done? Hmmm. x should be greater than 0 but smaller than 5, or smaller than 9?

Which do we pick? Right. We have to pick x < 5, because it's impossible to go any bigger.

Thus, our real world domain of V(x) is between 0 and 5.

Hope this helps! Remember to:

1) Read the problem

2) Look up concepts and definitions

3) Write what you know.

4) Write what you are looking to solve or answer.

5) Draw a picture

6) Ask if it makes sense.

7) Are you done?

Hope this helps!

Clint, Wyzant Tutor.

Hi Felicia! Great question there.

12p² + 8p² = ? 

One of the big ideas in algebra is 'like terms'. You may have seen or heard it. What does it mean? Basically, it comes down to apples and oranges. If you have 12 apples and 8 oranges, what do you have? It's a trick question. You can't really add them. You just have 12 apples and 8 oranges.You could add 3 apples to 8 apples or you could subtract 10 oranges from 18 oranges because they are the same type.

Im algebra we use letters and variables to keep track of types. We can add x's "5x + 3x = 8x" for example. You may have noticed the p^2 in the problem. That means "p squared"' or "p * p". You can add p^2 together just like x's or apples or oranges.

The steps are:

1) look at all the terms in the expression  "12p^2 + 8p^2 

We see there is just one type of term, and that's "p^2"

2) add or subtract the coefficients in front of the terms. A coefficient is thno umber in front of the term. They can be positive or negative. 

Here the coefficients are +12 and + 8

3) Find the sum of all the terms. That is your new coefficient. 

Here +12 +8 = 20. So our new coefficient is 20

4) Write the new coefficients next to their terms. Here we just have one term, p^2

=20p^2

Other expressions may have many terms.

Can you add p and p^2? They aren't like terms. Can you add x and y? Not like terms. Can you add x^3 and x^3?  Yes! Like terms!

Hope this helps

Clint

Hi Jackie,

It might help to think of functions as little machines or factories that take some input, do something to it, then give you the result. A function definition like h(x) = -2x + 5  tells us what our function, h, does to the input, x. Whatever is in the parentheses is the input. If you see h(4), that means we give 4 to our function h. h(4) = -2 * 4 + 5 = -8 + 5 = -3, so h(4) = -3. 

What if you wanted a function that doubled a number? That could be f(x) = 2x.  Does it work? Let's try x = 3. We know if you double 3 you get 6. Does f(3) = 6? f(3) = 2 * 3 = 6. So yes!

If you wanted a function that subtracted 3 from any number? That might be g(x) = x - 3.

The neat thing about functions is that they can work on numbers, variables, or even other functions. That's what your question is asking. If you understand that functions take input, do their thing, then give the result, then you might guess how to proceed.

f(x) = -x + 5       g(x) = 4x + 2

A. f(g(x))? This looks complicated, but it's asking what the function f does to the input g(x). That's why g(x) is in parentheses. We know what it does to x, or to a number by substituting. We just do the same with the whole function.

f(g(x)) = - g(x) + 5 = -(4x + 2) + 5 = -4x - 8 + 5 = -4x - 3

B. g(f(x)) is asking what? What does g(x) do to f(x) as input? Again, we substitute just like with the numbers.

g(f(x)) = 4 f(x) + 2 = 4 (-x + 5) + 2 = -4x + 20 + 2 = -4x + 22

When you run across functions and compositions of functions, just remember they are little machines and do their work one step at a time.