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Divide the polynomials 4x

4
+4x−10 by 2x
2
−3 to determine the quotient and remainder.
The quotient is □ and the remainder is □.

Options:
A. 2x
2
+3
B. 4x−1
C. 2x
2
−3
D. 4x−19

3 answers

To divide the polynomial \( 4x^4 + 4x - 10 \) by \( 2x^2 - 3 \), we can use polynomial long division.

  1. Divide the first term of the dividend by the first term of the divisor: \[ \frac{4x^4}{2x^2} = 2x^2 \] This is the first term of the quotient.

  2. Multiply the entire divisor by this term: \[ 2x^2(2x^2 - 3) = 4x^4 - 6x^2 \]

  3. Subtract this from the original polynomial: \[ (4x^4 + 4x - 10) - (4x^4 - 6x^2) = 6x^2 + 4x - 10 \]

  4. Now repeat the process with the new polynomial \( 6x^2 + 4x - 10 \): \[ \frac{6x^2}{2x^2} = 3 \] This is the next term of the quotient.

  5. Multiply the entire divisor by this new term: \[ 3(2x^2 - 3) = 6x^2 - 9 \]

  6. Subtract this result from \( 6x^2 + 4x - 10 \): \[ (6x^2 + 4x - 10) - (6x^2 - 9) = 4x - 1 \]

Now, since \( 4x - 1 \) has a degree less than \( 2x^2 - 3 \), we stop here.

The final result of the division is:

  • Quotient: \( 2x^2 + 3 \)
  • Remainder: \( 4x - 1 \)

Putting it all together, we find: \[ \text{The quotient is } 2x^2 + 3 \text{ and the remainder is } 4x - 1. \]

However, we need to check the options provided for the values.

Options given: A. \( 2x^2 + 3 \)
B. \( 4x - 1 \)
C. \( 2x^2 - 3 \)
D. \( 4x - 19 \)

Thus, the appropriate selections corresponding to the completed division are:

  • The quotient is \( 2x^2 + 3 \) (Option A).
  • The remainder is \( 4x - 1 \) (which is not directly listed as an option, but it corresponds to B when the representation is simplified).

Final output:

  • The quotient is 2x^2 + 3 and the remainder is 4x - 1.
Are you sure

Let's carefully go through the polynomial long division process again to ensure accuracy. We need to divide \( 4x^4 + 4x - 10 \) by \( 2x^2 - 3 \).

Step 1: Setup the Division

We start with:

Dividend: \[ 4x^4 + 0x^3 + 4x - 10 \]

Divisor: \[ 2x^2 - 3 \]

Step 2: Divide the Leading Terms

  1. Divide the leading term of the dividend by the leading term of the divisor: \[ \frac{4x^4}{2x^2} = 2x^2 \] So our first term in the quotient is \( 2x^2 \).

Step 3: Multiply and Subtract

  1. Multiply the entire divisor by \( 2x^2 \): \[ 2x^2(2x^2 - 3) = 4x^4 - 6x^2 \]
  2. Now subtract this from the original polynomial: \[ (4x^4 + 0x^3 + 4x - 10) - (4x^4 - 6x^2) = 0x^4 + 0x^3 + 6x^2 + 4x - 10 \] This simplifies to: \[ 6x^2 + 4x - 10 \]

Step 4: Repeat the Process

  1. Now we take \( 6x^2 + 4x - 10 \) and divide by \( 2x^2 - 3 \): \[ \frac{6x^2}{2x^2} = 3 \] Thus, our next term in the quotient is \( 3 \).

  2. Multiply the divisor by \( 3 \): \[ 3(2x^2 - 3) = 6x^2 - 9 \]

  3. Subtract this from \( 6x^2 + 4x - 10 \): \[ (6x^2 + 4x - 10) - (6x^2 - 9) = 4x - 1 \]

Final Result

We now have:

  • Quotient: \( 2x^2 + 3 \)
  • Remainder: \( 4x - 1 \)

After reviewing the options given:

  • The quotient is \( 2x^2 + 3 \), which corresponds to Option A.
  • The remainder \( 4x - 1 \) corresponds to Option B.

So, the conclusion remains that:

  • The quotient is \( 2x^2 + 3 \).
  • The remainder is \( 4x - 1 \).

I apologize if there was any confusion in the prior explanation, but the calculations and the resulting options are accurate. Thank you for your patience!