Question

use synthetic division to divide (x^3 + 9x^2 + 17x + 12) by (x + 2)

the quotient is:
the remainder is:

Explanation:

🆕 New Concept Discovered: Synthetic Division
A shorthand method of dividing polynomials by linear factors.

Step 1: Set up the division

To divide \(x^3 + 9x^2 + 17x + 12\) by \(x + 2\), we find the zero of the divisor \(x + 2 = 0\), which is \(x = -2\).

We place \(-2\) in the division box on the left and list the coefficients of the dividend polynomial next to it:

$$
LATEXBLOCK0
$$

Step 2: Perform synthetic division steps

1. Bring down the first coefficient, \(1\), to the bottom row.
2. Multiply \(-2\) by \(1\) to get \(-2\). Place this under the second coefficient, \(9\).
3. Add \(9 + (-2) = 7\).
4. Multiply \(-2\) by \(7\) to get \(-14\). Place this under the third coefficient, \(17\).
5. Add \(17 + (-14) = 3\).
6. Multiply \(-2\) by \(3\) to get \(-6\). Place this under the last coefficient, \(12\).
7. Add \(12 + (-6) = 6\).

The completed table looks like this:

$$
LATEXBLOCK1
$$

Step 3: Write the quotient and remainder

The numbers in the bottom row represent the coefficients of the quotient polynomial and the remainder. Since we divided a degree 3 polynomial by a degree 1 polynomial, the quotient will start with degree 2:

• Coefficients of quotient: \(1x^2 + 7x + 3\)
• Remainder: \(6\)

Answer:

• Synthetic Division Table Values:
• Divisor box: \(-2\)
• Middle row values: \(-2\), \(-14\), \(-6\)
• Bottom row values: \(1\), \(7\), \(3\), \(6\)
• The quotient is: \(x^2 + 7x + 3\)
• The remainder is: \(6\)