Use synthetic division to find the quotient and remainder if the first polynomial is divided by the second

 Use synthetic division to find the quotient and remainder if the first polynomial is divided by the second

The Correct Answer and Explanation is:

Of course. Here is the correct answer and a detailed explanation for the math problem.

quotient — 12x³ + 6x² – 8x – 4
remainder — 0

Explanation

The problem asks to use synthetic division to divide the polynomial 12x⁴ – 11x² + 2 by x – 1/2. Synthetic division is a simplified method of polynomial division, specifically used when the divisor is a linear expression of the form x – c.

Step 1: Set up the Division

First, identify the coefficients of the dividend polynomial, 12x⁴ – 11x² + 2. It’s crucial to list the coefficients for every power of x in descending order, from the highest power down to the constant term. If a term is missing, we must use a zero as a placeholder. In this case, the x³ and x terms are missing.

The polynomial can be rewritten as: 12x⁴ + 0x³ – 11x² + 0x + 2.
The coefficients are: 12, 0, -11, 0, 2.

Next, identify the value of c from the divisor, x – c. For the divisor x – 1/2, the value of c is 1/2.

We set up the synthetic division problem by writing c to the left and the coefficients of the dividend to the right:Generated code

1/2 | 12   0   -11   0   2
    |
    -----------------------

Step 2: Perform the Synthetic Division

1. Bring down the first coefficient (12) below the line.
2. Multiply this number (12) by c (1/2): 12 * (1/2) = 6. Write this result under the second coefficient (0).
3. Add the numbers in the second column: 0 + 6 = 6.
4. Repeat the process: multiply the new number (6) by c: 6 * (1/2) = 3. Write this under the third coefficient (-11) and add: -11 + 3 = -8.
5. Continue this pattern: -8 * (1/2) = -4. Write this under the next coefficient (0) and add: 0 + (-4) = -4.
6. For the final step: -4 * (1/2) = -2. Write this under the last coefficient (2) and add: 2 + (-2) = 0.

The completed division looks like this:Generated code

1/2 | 12   0   -11    0    2
    |      6      3   -4   -2
    --------------------------
      12   6     -8   -4    0

Step 3: Interpret the Result

The numbers in the bottom row represent the coefficients of the quotient and the remainder. The last number, 0, is the remainder. The other numbers, 12, 6, -8, -4, are the coefficients of the quotient polynomial.

The degree of the quotient is always one less than the degree of the dividend. Since the dividend was a 4th-degree polynomial, the quotient will be a 3rd-degree polynomial. Using the coefficients from our result, the quotient is 12x³ + 6x² – 8x – 4.