SOLUTION: Let f(x) = x^3 + 3x^2 + 4x - 7 and g(x) = 2x^4 - 8x^3 + 4x^2 - 1. What is the coefficient of x^2 in the sum polynomial f(x)^2*g(x)?

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Let f(x) = x^3 + 3x^2 + 4x - 7 and g(x) = 2x^4 - 8x^3 + 4x^2 - 1. What is the coefficient of x^2 in the sum polynomial f(x)^2*g(x)?      Log On


   

Question 1209335: Let f(x) = x^3 + 3x^2 + 4x - 7 and g(x) = 2x^4 - 8x^3 + 4x^2 - 1. What is the coefficient of x^2 in the sum polynomial f(x)^2*g(x)?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.
Let f(x) = x^3 + 3x^2 + 4x - 7 and g(x) = 2x^4 - 8x^3 + 4x^2 - 1. What is the coefficient of x^2 in the sum polynomial f(x)^2*g(x)?
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As worded, printed, presented in the post, the problem is written incorrectly (has error/errors),

is self-contradictory and makes no sense.

Double check your writing, find possible error/errors, fix it/them, then re-post to the forum.


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                How the problem is worded and printed in the post,
            it is worded and printed mathematically in illiterate way,
            and this illiteracy sticks out very much and very explicitly.

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Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

First we have to determine a few terms of f%5E2 i.e. %28+f%28x%29+%29%5E2

I'll use the box method.

%28+f%28x%29+%29%5E2 is the same as +f%28x%29+%2A+f%28x%29+
Write the terms of f(x) along the top row and along the left hand side
x^33x^24x-7
x^3
3x^2
4x
-7

We then fill out this table by multiplying the headers (eg: 7 times 7 = 49 in the bottom right corner)

We don't have to fill out the entire table.
Since we only want the coefficient x^2 at the end, we just need the terms that have exponent 2 or smaller.
x^33x^24x-7
x^3
3x^2-21x^2
4x16x^2-28x
-7-21x^2-28x49

Combining like terms gives -26x^2-56x+49

This demonstrates that +%28+f%28x%29+%29%5E2+=+h%28x%29+-26x%5E2-56x%2B49 where h(x) is some 6th degree polynomial and its last term is some cubic monomial.
We don't need to worry about h(x) since it won't influence the x^2 coefficient at the end.

We'll use the box method again.
Since h(x) doesn't play a role, we can simply ignore it to focus on the -26x^2-56x+49 portion.
Write those terms along the top row and the terms of g(x) along the left column.
-26x^2-56x49
2x^4
-8x^3
4x^2196x^2
-126x^2

Any cell that's blank won't play a role in the final answer.
The terms that do contribute a role are the 26x^2 and 196x^2.
Those add to 222x^2


If you wanted, you can expand out every term of f%28x%29%2Af%28x%29%2Ag%28x%29 to get

but it would take a bit longer than the previous method discussed above.
Or you can use an online calculator such as GeoGebra, WolframAlpha, etc to quickly arrive at that massive 10th degree polynomial.
I recommend to use such a specialized calculator only if you are checking your answer.


Answer: 222