Asked by cru
Use long division to divide the polynomials: x3βx2+4x+2%0D%0Aπ₯%0D%0A3%0D%0Aβ%0D%0Aπ₯%0D%0A2%0D%0A+%0D%0A4%0D%0Aπ₯%0D%0A+%0D%0A2%0D%0A.(1 point)%0D%0AResponses%0D%0A%0D%0Ax2β3x+6β8x+2%0D%0Aπ₯%0D%0A2%0D%0Aβ%0D%0A3%0D%0Aπ₯%0D%0A+%0D%0A6%0D%0Aβ%0D%0A8%0D%0Aπ₯%0D%0A+%0D%0A2%0D%0Ax squared minus 3 x plus 6 minus Start Fraction 8 over x plus 2 End Fraction%0D%0A%0D%0Ax2β3x+6+6x+2%0D%0Aπ₯%0D%0A2%0D%0Aβ%0D%0A3%0D%0Aπ₯%0D%0A+%0D%0A6%0D%0A+%0D%0A6%0D%0Aπ₯%0D%0A+%0D%0A2%0D%0Ax squared minus 3 x plus 6 plus Start Fraction 6 over x plus 2 End Fraction%0D%0A%0D%0Ax2+3x+6β8x+2%0D%0Aπ₯%0D%0A2%0D%0A+%0D%0A3%0D%0Aπ₯%0D%0A+%0D%0A6%0D%0Aβ%0D%0A8%0D%0Aπ₯%0D%0A+%0D%0A2%0D%0Ax squared plus 3 x plus 6 minus Start Fraction 8 over x plus 2 End Fraction%0D%0A%0D%0Ax2β2x+6β6x+2%0D%0Aπ₯%0D%0A2%0D%0Aβ%0D%0A2%0D%0Aπ₯%0D%0A+%0D%0A6%0D%0Aβ%0D%0A6%0D%0Aπ₯%0D%0A+%0D%0A2%0D%0Ax squared minus 2 x plus 6 minus Start Fraction 6 over x plus 2 End Fraction
Answers
Answered by
GPT-4o mini
To perform the long division of polynomials to divide \( x^3 - x^2 + 4x + 2 \) by \( x \), we can follow these steps:
1. Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \), which gives us \( x^2 \).
2. Multiply the entire divisor \( x \) by \( x^2 \) to get \( x^3 \).
3. Subtract this result from the original polynomial:
\[
(x^3 - x^2 + 4x + 2) - (x^3) = -x^2 + 4x + 2
\]
4. Now repeat the process for the new polynomial \(-x^2 + 4x + 2\). Divide the leading term \(-x^2\) by \(x\), which gives \(-x\).
5. Multiply the divisor by \(-x\) to get \(-x^2\), then subtract:
\[
(-x^2 + 4x + 2) - (-x^2) = 4x + 2
\]
6. Now repeat the process for \(4x + 2\). Divide \(4x\) by \(x\), which gives us \(4\).
7. Multiply the divisor \(x\) by \(4\) to get \(4x\):
\[
(4x + 2) - (4x) = 2
\]
Now what we have is:
\[
x^2 - x + 4 \text{ with a remainder of } 2.
\]
So we can write:
\[
\frac{x^3 - x^2 + 4x + 2}{x} = x^2 - x + 4 + \frac{2}{x}
\]
Now let's match the response with the original options given:
The division result can be summarized as:
\[
x^2 - x + 4 + \frac{2}{x}
\]
Here are the simplified terms put together for better clarity, and the correct response from the options you provided is:
**Answer:**
None of the options presented closely matches the division result of \( x^2 - x + 4 + \frac{2}{x} \).
1. Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \), which gives us \( x^2 \).
2. Multiply the entire divisor \( x \) by \( x^2 \) to get \( x^3 \).
3. Subtract this result from the original polynomial:
\[
(x^3 - x^2 + 4x + 2) - (x^3) = -x^2 + 4x + 2
\]
4. Now repeat the process for the new polynomial \(-x^2 + 4x + 2\). Divide the leading term \(-x^2\) by \(x\), which gives \(-x\).
5. Multiply the divisor by \(-x\) to get \(-x^2\), then subtract:
\[
(-x^2 + 4x + 2) - (-x^2) = 4x + 2
\]
6. Now repeat the process for \(4x + 2\). Divide \(4x\) by \(x\), which gives us \(4\).
7. Multiply the divisor \(x\) by \(4\) to get \(4x\):
\[
(4x + 2) - (4x) = 2
\]
Now what we have is:
\[
x^2 - x + 4 \text{ with a remainder of } 2.
\]
So we can write:
\[
\frac{x^3 - x^2 + 4x + 2}{x} = x^2 - x + 4 + \frac{2}{x}
\]
Now let's match the response with the original options given:
The division result can be summarized as:
\[
x^2 - x + 4 + \frac{2}{x}
\]
Here are the simplified terms put together for better clarity, and the correct response from the options you provided is:
**Answer:**
None of the options presented closely matches the division result of \( x^2 - x + 4 + \frac{2}{x} \).