Question
What is the product of the following two polynomials (x^2 +2x) (3x^3 -4x^2 +x)?
a) 3x^5 -4x^4 +x^3 +6x -8x^3 +2x^2
b)3x^5 -4x^4 +7x^3 -8x^2
c)3x^5 +2x^4 -4x^3 +6x^3 -8x^2 +2x^2
d)3x^5 +6x^4 -8x^3 +2x^2
a) 3x^5 -4x^4 +x^3 +6x -8x^3 +2x^2
b)3x^5 -4x^4 +7x^3 -8x^2
c)3x^5 +2x^4 -4x^3 +6x^3 -8x^2 +2x^2
d)3x^5 +6x^4 -8x^3 +2x^2
Answers
GPT-4o mini
To find the product of the two polynomials \((x^2 + 2x)(3x^3 - 4x^2 + x)\), we can distribute each term in the first polynomial across each term in the second polynomial.
Starting with \(x^2\):
\[
x^2 \cdot 3x^3 = 3x^5
\]
\[
x^2 \cdot (-4x^2) = -4x^4
\]
\[
x^2 \cdot x = x^3
\]
Now for \(2x\):
\[
2x \cdot 3x^3 = 6x^4
\]
\[
2x \cdot (-4x^2) = -8x^3
\]
\[
2x \cdot x = 2x^2
\]
Now, we combine all of these results:
\[
3x^5 + (-4x^4 + 6x^4) + (x^3 - 8x^3) + 2x^2
\]
This simplifies to:
\[
3x^5 + (6x^4 - 4x^4) + (1x^3 - 8x^3) + 2x^2
\]
\[
= 3x^5 + 2x^4 - 7x^3 + 2x^2
\]
Upon reviewing the options given:
- a) \(3x^5 - 4x^4 + x^3 + 6x - 8x^3 + 2x^2\)
- b) \(3x^5 - 4x^4 + 7x^3 - 8x^2\)
- c) \(3x^5 + 2x^4 - 4x^3 + 6x^3 - 8x^2 + 2x^2\)
- d) \(3x^5 + 6x^4 - 8x^3 + 2x^2\)
None of these seemed to match our derived equation, but let's refine and check the coefficients carefully.
Combining terms correctly will yield:
\[
= 3x^5 + (6x^4 - 4x^4) + (1x^3 - 8x^3) + 2x^2 = 3x^5 + 2x^4 - 7x^3 + 2x^2
\]
So even after careful re-evaluating of all derivatives and simplifications, the direct comparisons do not match any given choices.
Lets check choice c:
\[
3x^5 + 2x^4 - 4x^3 + 6x^3 - 8x^2 + 2x^2 = 3x^5 + 2x^4 + (6-4)x^3 + (-8 + 2)x^2
\]
\[
= 3x^5 + 2x^4 + 2x^3 - 6x^2
\]
This is still distinct enough to set aside the original result. Therefore, from the availability, no stated choice matched the produced resultant polynomial of \[3x^5 + 2x^4 - 7x^3 + 2x^2\].
Thus, none of the provided options are accurate for the polynomial multiplication calculated. Please verify given polynomials or investigate for any possible input contradictions!
Starting with \(x^2\):
\[
x^2 \cdot 3x^3 = 3x^5
\]
\[
x^2 \cdot (-4x^2) = -4x^4
\]
\[
x^2 \cdot x = x^3
\]
Now for \(2x\):
\[
2x \cdot 3x^3 = 6x^4
\]
\[
2x \cdot (-4x^2) = -8x^3
\]
\[
2x \cdot x = 2x^2
\]
Now, we combine all of these results:
\[
3x^5 + (-4x^4 + 6x^4) + (x^3 - 8x^3) + 2x^2
\]
This simplifies to:
\[
3x^5 + (6x^4 - 4x^4) + (1x^3 - 8x^3) + 2x^2
\]
\[
= 3x^5 + 2x^4 - 7x^3 + 2x^2
\]
Upon reviewing the options given:
- a) \(3x^5 - 4x^4 + x^3 + 6x - 8x^3 + 2x^2\)
- b) \(3x^5 - 4x^4 + 7x^3 - 8x^2\)
- c) \(3x^5 + 2x^4 - 4x^3 + 6x^3 - 8x^2 + 2x^2\)
- d) \(3x^5 + 6x^4 - 8x^3 + 2x^2\)
None of these seemed to match our derived equation, but let's refine and check the coefficients carefully.
Combining terms correctly will yield:
\[
= 3x^5 + (6x^4 - 4x^4) + (1x^3 - 8x^3) + 2x^2 = 3x^5 + 2x^4 - 7x^3 + 2x^2
\]
So even after careful re-evaluating of all derivatives and simplifications, the direct comparisons do not match any given choices.
Lets check choice c:
\[
3x^5 + 2x^4 - 4x^3 + 6x^3 - 8x^2 + 2x^2 = 3x^5 + 2x^4 + (6-4)x^3 + (-8 + 2)x^2
\]
\[
= 3x^5 + 2x^4 + 2x^3 - 6x^2
\]
This is still distinct enough to set aside the original result. Therefore, from the availability, no stated choice matched the produced resultant polynomial of \[3x^5 + 2x^4 - 7x^3 + 2x^2\].
Thus, none of the provided options are accurate for the polynomial multiplication calculated. Please verify given polynomials or investigate for any possible input contradictions!