Question
2x^(3)+3x^(2)-21x+20 divided by x-2 if there is a remainder express the result is the form of q(x)+(r(x))/(b(x)) explain simply and without all the brackets pleae
Answers
Answered by
GPT-4o mini
To divide the polynomial \(2x^3 + 3x^2 - 21x + 20\) by \(x - 2\), we can use polynomial long division.
1. **Setup the Division**:
We want to divide \(2x^3 + 3x^2 - 21x + 20\) by \(x - 2\).
2. **Divide the Leading Terms**:
Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x\):
\[
2x^2
\]
This will be the first term of the quotient.
3. **Multiply and Subtract**:
Multiply \(2x^2\) by \(x - 2\):
\[
2x^2(x - 2) = 2x^3 - 4x^2
\]
Now subtract this from the original polynomial:
\[
(2x^3 + 3x^2) - (2x^3 - 4x^2) = 3x^2 + 4x^2 = 7x^2
\]
Bring down the next term, which is \(-21x\):
\[
7x^2 - 21x
\]
4. **Repeat the Process**:
Now divide \(7x^2\) by \(x\):
\[
7x
\]
Multiply \(7x\) by \(x - 2\):
\[
7x(x - 2) = 7x^2 - 14x
\]
Subtract:
\[
(7x^2 - 21x) - (7x^2 - 14x) = -21x + 14x = -7x
\]
Bring down the next term (constant 20):
\[
-7x + 20
\]
5. **Divide Again**:
Now divide \(-7x\) by \(x\):
\[
-7
\]
Multiply \(-7\) by \(x - 2\):
\[
-7(x - 2) = -7x + 14
\]
Subtract:
\[
(-7x + 20) - (-7x + 14) = 20 - 14 = 6
\]
6. **Final Result**:
Now we have determined the quotient and the remainder:
- Quotient \(q(x) = 2x^2 + 7x - 7\)
- Remainder \(r(x) = 6\)
Putting it all together, we can express the result as:
\[
2x^2 + 7x - 7 + \frac{6}{x - 2}
\]
So, in the requested form, it's:
\[
q(x) + \frac{r(x)}{b(x)} = 2x^2 + 7x - 7 + \frac{6}{x - 2}
\]
1. **Setup the Division**:
We want to divide \(2x^3 + 3x^2 - 21x + 20\) by \(x - 2\).
2. **Divide the Leading Terms**:
Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x\):
\[
2x^2
\]
This will be the first term of the quotient.
3. **Multiply and Subtract**:
Multiply \(2x^2\) by \(x - 2\):
\[
2x^2(x - 2) = 2x^3 - 4x^2
\]
Now subtract this from the original polynomial:
\[
(2x^3 + 3x^2) - (2x^3 - 4x^2) = 3x^2 + 4x^2 = 7x^2
\]
Bring down the next term, which is \(-21x\):
\[
7x^2 - 21x
\]
4. **Repeat the Process**:
Now divide \(7x^2\) by \(x\):
\[
7x
\]
Multiply \(7x\) by \(x - 2\):
\[
7x(x - 2) = 7x^2 - 14x
\]
Subtract:
\[
(7x^2 - 21x) - (7x^2 - 14x) = -21x + 14x = -7x
\]
Bring down the next term (constant 20):
\[
-7x + 20
\]
5. **Divide Again**:
Now divide \(-7x\) by \(x\):
\[
-7
\]
Multiply \(-7\) by \(x - 2\):
\[
-7(x - 2) = -7x + 14
\]
Subtract:
\[
(-7x + 20) - (-7x + 14) = 20 - 14 = 6
\]
6. **Final Result**:
Now we have determined the quotient and the remainder:
- Quotient \(q(x) = 2x^2 + 7x - 7\)
- Remainder \(r(x) = 6\)
Putting it all together, we can express the result as:
\[
2x^2 + 7x - 7 + \frac{6}{x - 2}
\]
So, in the requested form, it's:
\[
q(x) + \frac{r(x)}{b(x)} = 2x^2 + 7x - 7 + \frac{6}{x - 2}
\]
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