Question
Divide the polynomials
4x^2 + 4x - 10 by 2x^2 - 3
to determine the quotient and remainder.
A
The quotient is 2x2−32x^2-32x
2
−3. The remainder is 4x−194x-194x−19.
B
The quotient is 2x2+32x^2+32x
2
+3. The remainder is 4x−14x-14x−1.
C
The quotient is 2x2−32x^2-32x
2
−3. The remainder is 4x+14x+14x+1.
D
The quotient is 2x2+32x^2+32x
2
+3. The remainder is 4x−194x-194x−19.
Answers
Answered by
GPT-4o mini
To divide the polynomial \( 4x^2 + 4x - 10 \) by \( 2x^2 - 3 \), we can follow polynomial long division. Let's do this step by step:
1. **Divide the leading terms**:
- Divide \( 4x^2 \) by \( 2x^2 \) to get \( 2 \).
2. **Multiply**:
- Multiply \( 2 \) by \( 2x^2 - 3 \) to get \( 4x^2 - 6 \).
3. **Subtract**:
- Now subtract \( (4x^2 - 6) \) from \( (4x^2 + 4x - 10) \):
\[
(4x^2 + 4x - 10) - (4x^2 - 6) = 4x - 10 + 6 = 4x - 4.
\]
4. **Now we have the new polynomial**:
- The result of the subtraction gives us \( 4x - 4 \).
5. **Now divide \( 4x - 4 \) by \( 2x^2 - 3 \)**:
- Since the degree of \( 4x - 4 \) (which is 1) is less than the degree of \( 2x^2 - 3 \) (which is 2), we cannot divide any further.
6. **Final Result**:
- Thus, the quotient is \( 2 \) and the remainder is \( 4x - 4 \).
Therefore, the result of the division is:
- Quotient: \( 2 \)
- Remainder: \( 4x - 4 \)
Now we need to match it with the given options:
- The expressions in the answer choices can be misleading. Let's confirm which option it matches:
None of the provided options A, B, C, or D accurately represent a quotient of \( 2 \) with a remainder of \( 4x - 4 \).
If we represent \( 4x - 4 \) clearly:
- \( 4x - 4 = 4(x - 1) \).
Thus the correct quotient and remainder based on our calculation should be:
- Quotient: \( 2 \).
- Remainder: \( 4x - 4 \).
Please double-check the given options for typographical errors or reformulations.
1. **Divide the leading terms**:
- Divide \( 4x^2 \) by \( 2x^2 \) to get \( 2 \).
2. **Multiply**:
- Multiply \( 2 \) by \( 2x^2 - 3 \) to get \( 4x^2 - 6 \).
3. **Subtract**:
- Now subtract \( (4x^2 - 6) \) from \( (4x^2 + 4x - 10) \):
\[
(4x^2 + 4x - 10) - (4x^2 - 6) = 4x - 10 + 6 = 4x - 4.
\]
4. **Now we have the new polynomial**:
- The result of the subtraction gives us \( 4x - 4 \).
5. **Now divide \( 4x - 4 \) by \( 2x^2 - 3 \)**:
- Since the degree of \( 4x - 4 \) (which is 1) is less than the degree of \( 2x^2 - 3 \) (which is 2), we cannot divide any further.
6. **Final Result**:
- Thus, the quotient is \( 2 \) and the remainder is \( 4x - 4 \).
Therefore, the result of the division is:
- Quotient: \( 2 \)
- Remainder: \( 4x - 4 \)
Now we need to match it with the given options:
- The expressions in the answer choices can be misleading. Let's confirm which option it matches:
None of the provided options A, B, C, or D accurately represent a quotient of \( 2 \) with a remainder of \( 4x - 4 \).
If we represent \( 4x - 4 \) clearly:
- \( 4x - 4 = 4(x - 1) \).
Thus the correct quotient and remainder based on our calculation should be:
- Quotient: \( 2 \).
- Remainder: \( 4x - 4 \).
Please double-check the given options for typographical errors or reformulations.