Quiz General quiz on polynomials

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Question: I would REALLY appreciate it if a tutor could assist me with the following problem:
Find all the asymptotes of f(x) = +%282x%5E2-3x-5%29+%2F+%28x%5E2-7x%2B12%29+
I believe there is an oblique and possibly a vertical asymptote.
Thanks so much!!!
Answers:

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I don't think this has an oblique asymptote. It definitely has a horizontal (<--->) asymptote, though. The way to tell that is to look at the terms with the highest power in both numerator and denominator. We've got the 2x^2 in the numerator and the x^2 in the denominator. You simply just take the numerator and divide by the denominator, and you'll get 2. The horizontal asymptote occurs at y = 2. BTW, the highest powers in both numerator AND denominator HAVE TO MATCH POWERS!

What about the horizontal asymptotes? That's when you have to play around with the denominator. There are "troublemaker" values that you can't plug in to this function because they will make the denominator zero, which you know can't happen. So we'll have to find those troublemaker values that make the denominator zero.

The denominator is +x%5E2+-+7x+%2B+12+. So you ask yourself "what times what equals 12 and their sum or diference must come out to -7?". After a while, you'll come up with -3 and -4. So then you write (x - 3)(x - 4). The values 3 and 4 will make the denominator zero, so you CAN'T plug those into your function. Graphically, this gives you the vertical asymptotes x=3 and x=4.

I don't remember the rules for having an oblique asymptote, though. If I can remember correctly, an oblique happens when the term with the highest power in the numerator is of a higher power than the denominator's highest power term.
Question: Simplify 7x%5E3z%5E5%2F4z%5E15
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7x%5E3z%5E5%2F4z%5E15

simplify
7x%5E3%2F4z%5E10

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7x^3/4z^10
Question: how do i solve this problem
(x-1)^2(x+3)^2=
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equal what? you have an equals sign with nothing on the right hand side.

You are doing maths here, and the one thing you have to be is precise/exact in your question.

Do you want a solution of just to simplify the polynomial? Or something else perhaps?

Jon.
Question: How would I go about simplifying this problem.
x^2-x-6 / x^2-9
Thank You for your help.
Answers:

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Factorise the 2 expressions: %28x%5E2-x-6%29%2F%28x%5E2-9%29 and you should be able to cancel some of the things then...that is the aim. If we cannot, then i would suggest that we have done it wrong and then check your working.

%28%28x-3%29%28x%2B2%29%29%2F%28%28x%2B3%29%28x-3%29%29
%28x%2B2%29%2F%28x%2B3%29

jon.
Question: Can you please help me answer this equaton?
Combine:(3x^3-4x^2)+(-7-3x^3)-(-4x^2+2x)
Answers:

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Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • Graphical form: %283x%5E3-4x%5E2%29%2B%28-7-3x%5E3%29-%28-4x%5E2%2B2x%29 simplifies to -2%2Ax-7
  • Text form: (3x^3-4x^2)+(-7-3x^3)-(-4x^2+2x) simplifies to -2*x-7
  • Cartoon (animation) form: simplify_cartoon%28+%283x%5E3-4x%5E2%29%2B%28-7-3x%5E3%29-%28-4x%5E2%2B2x%29+%29
    For tutors: simplify_cartoon( (3x^3-4x^2)+(-7-3x^3)-(-4x^2+2x) )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at highlight_red%28+%283%2Ax%5E3-4%2Ax%5E2%29+%29%2B%28-7-3%2Ax%5E3%29-%28-4%2Ax%5E2%2B2%2Ax%29.
Remove unneeded parentheses around terms highlight_red%28+3%2Ax%5E3+%29,highlight_red%28+-4%2Ax%5E2+%29
It becomes .
Look at 3%2Ax%5E3-4%2Ax%5E2%2Bhighlight_red%28+%28-7-3%2Ax%5E3%29+%29-%28-4%2Ax%5E2%2B2%2Ax%29.
Remove unneeded parentheses around terms highlight_red%28+-7+%29,highlight_red%28+-3%2Ax%5E3+%29
It becomes .
Look at 3%2Ax%5E3-4%2Ax%5E2-7-3%2Ax%5E3-highlight_red%28+%28-4%2Ax%5E2%2B2%2Ax%29+%29.
Remove unneeded parentheses around terms highlight_red%28+4%2Ax%5E2+%29,highlight_red%28+-2%2Ax+%29
It becomes .
Look at 3%2Ax%5E3-4%2Ax%5E2-highlight_red%28+7+%29-3%2Ax%5E3%2B4%2Ax%5E2-2%2Ax.
Moved -7 to the right of expression
It becomes 3%2Ax%5E3-4%2Ax%5E2-3%2Ax%5E3%2B4%2Ax%5E2-2%2Ax-highlight_green%28+7+%29.
Look at highlight_red%28+3%2Ax%5E3+%29-4%2Ax%5E2-highlight_red%28+3%2Ax%5E3+%29%2B4%2Ax%5E2-2%2Ax-7.
Eliminated similar terms highlight_red%28+3%2Ax%5E3+%29,highlight_red%28+-3%2Ax%5E3+%29 replacing them with highlight_green%28+%283-3%29%2Ax%5E3+%29
It becomes highlight_green%28+%283-3%29%2Ax%5E3+%29-4%2Ax%5E2%2B4%2Ax%5E2-2%2Ax-7.
Look at %283-3%29%2Ax%5E3-highlight_red%28+4%2Ax%5E2+%29%2Bhighlight_red%28+4%2Ax%5E2+%29-2%2Ax-7.
Eliminated similar terms highlight_red%28+-4%2Ax%5E2+%29,highlight_red%28+4%2Ax%5E2+%29 replacing them with highlight_green%28+%28-4%2B4%29%2Ax%5E2+%29
It becomes %283-3%29%2Ax%5E3%2Bhighlight_green%28+%28-4%2B4%29%2Ax%5E2+%29-2%2Ax-7.
Look at highlight_red%28+%283-3%29%2Ax%5E3+%29%2B%28-4%2B4%29%2Ax%5E2-2%2Ax-7.
Since highlight_red%28+%283-3%29%2Ax%5E3+%29 has zero as a factor, it should be replaced with a zero
Look at 0%2B%28-4%2B4%29%2Ax%5E2-2%2Ax-7.
Added fractions or integers together
It becomes 0%2B%28-4%2B4%29%2Ax%5E2-2%2Ax-7.
Look at 0%2Bhighlight_red%28+%28-4%2B4%29%2Ax%5E2+%29-2%2Ax-7.
Since highlight_red%28+%28-4%2B4%29%2Ax%5E2+%29 has zero as a factor, it should be replaced with a zero
Look at 0%2B0-2%2Ax-7.
Added fractions or integers together
It becomes 0%2B0-2%2Ax-7.
Look at highlight_red%28+0+%29%2B0-2%2Ax-7.
Remove extraneous zero highlight_red%28+0+%29
It becomes 0-2%2Ax-7.
Look at highlight_red%28+0+%29-2%2Ax-7.
Remove extraneous zero highlight_red%28+0+%29
It becomes -2%2Ax-7.
Result: -2%2Ax-7

Universal Simplifier and Solver


Done!
Question: How do I simplify %28%28x%5E2-2x-15%29%2F%28%28x-6%29%28x-3%29%29%29%2A%28%282x-6%29%2F%28x-5%29%29
Answers:

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Try this:
%28%28x%5E2-2x-15%29%2F%28%28x-6%29%28x-3%29%29%29%2A%28%282x-6%29%2F%28x-5%29%29
Factor the numerator of the 1st fraction.
%28%28%28x%2B3%29%28x-5%29%29%2F%28%28x-6%29%28x-3%29%29%29%2A%28%282x-6%29%2F%28x-5%29%29
Now factor 2 from the numerator of the 2nd fraction.
%28%28%28x%2B3%29%28x-5%29%29%2F%28%28x-6%29%28x-3%29%29%29%2A%282%28x-3%29%2F%28x-5%29%29
After cancelling like factors in top and bottom you are left with:
%282%28x%2B3%29%29%2F%28x-6%29
Does this help?
Question: please help me with this:

For a value of K, the expression x+k/x^2+kx-15 is undefined when x=5 and x=-3. Find the value of K
Answers:

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The trick here lies in the denominator. Hang on for the ride. This is a bit tricky.

I take it that you know that 5 and -3 will make the expression fail because those two values will force the denominator to be equal to 0.

They already gave you the fact that the denominator is quadratic (because of the x^2 term), and that quadratic expression can be factored in (x + _)(x + _) form (reverse-FOILING). We now need to set that denominator equal to zero to find which values of x make that denominator zero. In other words, we are looking for illegal values of x. So far, we have the equation

+%28x+%2B+_%29%28x+%2B+_%29+=+0+

But what values do you put in place of the blanks? Ah! They gave you the x-values already - the 5 and the -3. So, substitute the 5 and the -3 for each of the x's, NOT the blanks. Now we have

+%285+%2B+_%29%28-3+%2B+_%29+=+0+

Remember that 0 times anything equals zero. So we'll pick at the first set of parentheses and set it equal to zero:

+%285+%2B+_%29+=+0+ You'll see that _ = -5

Let's take the second set of parentheses and set it equal to zero.

+%28-3+%2B+_%29+=+0+ You'll see that _ = 3

Now rewrite the equation with the x's and place the two values you found for the blanks:

+%28x+-+5%29%28x+%2B+3%29+=+0+

NOTE: We could've just agreed to write write the quadratic in (x + _)(x + _) form and just plug in the undefined values given and change the signs. So we could've straight up said (x - 5)(x + 3) = 0 (plug in the numbers but reverse the signs just because it works out that way), but I just wanted to show you why and how it works that way.

Now, do FOIL on the (x - 5)(x + 3). This turns out to be +x%5E2+-+2x+-+15+. Now put that head to head with +x%5E2+%2B+kx+-+15+ and you'll see that k = -2.

WARNING. Don't let the "+ k" vs. the "- 2" fool you into believing that there is no way to work out this problem. When you see "+ k" or "+ any variable", it just means that k can be any number. If k turned out to be negative, like in this case, it really is "+ -2" which turns out to be "- 2".
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